I thought this was an interesting thread. I read through it carefully and thought I would try to add my own thoughts.
Firstly, I would like to look at quotes. These quotes come from the games or books themselves.
Let's start with Hyrule Historia. Here are some quotes:
"However, the destruction of Demise was not the end of the battle. It was the beginning of a curse: a never-ending cycle of the reincarnation of the Demon King, whose hatred for those with the blood of the Goddess and the spirit of the Hero is everlasting."-pg. 76
"The Hylians, descendants of Hylia, lived in Hyrule and possessed magical abilities." -pg 77
"The descendants of the goddess Hylia, who was reincarnated as Princess Zelda, established the kingdom of Hyrule and became Hyrule's royal family.... Many members of the royal family were born with special powers because of the lineage that connected them to the goddess. Princesses were repeatedly given the name Zelda, a name that came from historical legends." -pg 77.
"Until the era of the Hero of Time, the kingdom of Hyrule was mostly inhabited by the Hylian people.... However, the kingdom fell to ruin during the Imprisoning War, and the bloodline of the Hylians weakened with the ages, until their existence was naught but a thing of the past." -pg. 93
"Princess Zelda. In her veins flows the blood of the goddess Hylia." -pg. 93 (Link to the Past)
"The princess of Hyrule, and one of the maidens whose veins carry the blood of the sages. For generations, her line has possessed a mysterious power." - pg. 94 (Link to the Past)
"When she was little, Princess Zelda of Hyrule was chosen by the gods to wield divine power." -pg. 85 (Ocarina of Time)
"The princess of Hyrule, said to have mysterious powers." - pg. 79 (Minish Cap)
"One great king of Hyrule who had put the Triforce's power to use was concerned that its might would be abused after his death.... The monarch's magic ensured that a crest would appear upon a worthy person...." -pg. 104 (AoL)
"Princess Zelda, in whose veins flow the blood of the Hylians, possesses their characteristic long ears." -pg. 125 (WW)
"The Princess Zelda spoken of here is the fifth of her line, the great-great-grandchild of Tetra." - pg. 133 (ST)
"The mysterious power of Princess Zelda, along with Link's courage, brought about the end of Chancellor Cole and Malladus." - pg. 135 (ST)
So these are what is presented to us as fans about Zelda, her bloodline, and the powers of Hylians and the royal family. Let's do a little bit of interpreting now.
Early on we are told about Demise's curse, and that he would continually reincarnate to haunt future Zeldas and Links. As it has been pointed out, before, though, the translation of this curse has been debated and what it actually means. After all, this curse has failed at least twice in known Hyrulean history, and there are some demons that never mess with Princess Zelda at all. But I'm not really here to discuss that part.
From the remaining text, it is stated that Hylians are descendants from Hylia, with magical powers and pointed ears. It has been pointed out that pointy-ears Hylians existed prior to Hylia's reincarnation... but this is not entirely true. In SS, the word "Hylian" only exists in the form of the "Hylian Shield". And even then, in the Japanese version of the game, the "Hylian Shield" is actually called "Hylia Shield". So you could argue that "Hylian" never appears at all. They are instead just "humans" in SS, and would be known as Hylians in the far future. In MC, the word "Hylian" only appears once- "Hylian lore". Beyond that, it's still just "humans". It would not be until OoT when the word "human" vanishes and it becomes "Hylian" instead.
It could be argued, then, that early humans in SS were not "Hylians" because they did not have Hylia's blood yet. After all, HH does state that Hylians are descendants of Hylia. In the Japanese version, it says they share "roots" with the Goddess Hylia, in the way that she is their ancestor. NOTE: Where is says Hylia was reincarnated as Princess Zelda from page 77, this was a translation error. It just says "Zelda" originally, no princess title at all, just like in SS.
This would make sense. Logistically, Hylia's blood would spread throughout the population. It would take a few generations before her genetics were in every human, but it would eventually happen.
HH also points out that many royal family members have special powers due to their connection to the Goddess Hylia's bloodline. We see this not just with the princesses, but with the kings and princes as well. Several kings have demonstrated magical abilities either in-game or in the lore in general (King of Red Lions has powers, and the King from AoL casted an entire ancient spell that lasted beyond his death). But the princesses seem special. This isn't truly expanded upon in HH but we will get to that later. The bloodline doesn't need a female all the time though- I doubt AoL Princess Zelda ever had children before her slumber, so Princess Zelda from LoZ is likely descended from Princess Zelda I's older brother. This is speculation, but it tracks with the prince becoming King. Pretty sure his children would have inherited the throne first.
The issue here, however, is why the Royal Family is special. Why specifically them? Why do they get special powers but normal Hylians with the same ancestry do not? This is where I think we need to let go of logic a little bit. This IS a world of magic. There are spells, new bloodlines that pop up, souls that can wonder the earth and fairies that scream in your ear. Sometimes, it's best to let a game be a game.
But if we must apply logic, there are some ideas that come to mind. One is the possibility is that many Hylians may have had these special powers, but since the Royal Family kept such vast secrets (like the existence of the Triforce or secret tribes), they may have kept the powers close to their chest. We know from BotW that it is possible to not have an innate connection to one's powers. The Royal Family itself is not always perfect and free from doing dark deeds- Shadow Temple comes to mind, as well as the ancient king from Botw, and the Prince from AoL's backstory. If this is true, the knowledge of how to access such powers may have kept Hylians from using it in general, despite having a general knack for magic in general. This could also track with how Botw presents the powers- only Zelda's mother knew how to train Zelda, so while those who can access the powers easily, like Tetra, likely didn't need such training, if the training was isolated to just the Royal Family, it could have become forgotten knowledge in the common folk.
Another possibility is the way it is inherited. Since it IS magic, there could simply be a magical component that keeps these powers isolated to a few specific individuals. This does track with Botw as well, but more on that later.
Because we know from several games that the bloodline of the Hylians declines. LttP talks about them as a lost people, with very few remaining true Hylians left (Zelda, the maidens, and Link come to mind). In WW, most of the humans are round-eared, and those with Hylian blood have pointy ears. In TP, there are humans with round-ears as well. Creating a Champion makes mention that Gerudo women have likely gained their pointy ears due to many years of breeding with Hylian men, while in OoT the Gerudo still have rounded ears. All this to say that Hylian blood can wax and wane in groups of people. If Zelda only ever had children with another Hylian, while maybe her sister Catherine had children with a round-eared human, then Zelda's children would have a stronger connect to Hylia. Connections to the bloodline of the Goddess in many people, then, would have become lost. This would have isolated the bloodline to the Royal Family even more.
But now I want to quickly talk about the Princesses. As we can see from the quotes above, many of them mention that they are related to Hylia, just like any other Royal Family member. But I want to note that not all the Zelda's are equal. MC Zelda, for instance, is stated to have a mysterious power. This could be equated to Goddess powers. However, it could also be related to the Light Force, an artifact that lives inside her that was bestowed by the Minish. OoT Zelda is stated to have been chosen by the gods as a young age to wield divine powers. In other words, she has powers beyond her own bloodline- she was specifically chosen. LttP Zelda not only has Goddess blood, but also the blood of Sages, not something MC Zelda possesses. Some Zeldas have not presented any magical ability at all- LoZ Zelda comes to mind. That's not to say she didn't have powers, but they are never demonstrated or stated.
With ALL that said, I do want to jump to Creating a Champion and it's important quotes relating to Hylians and Hylia's bloodline:
"Zelda was the princess of the kingdom of Hyrule one hundred years prior to the events of Breath of the Wild and inherited the blood of the Goddess Hylia from her ancestors." -pg. 67.
"She agonizes over the fact that the sacred power that manifests in the women of the royal bloodline has not yet awakened in her..." -pg. 73
"Legend has it that the Hylian royal family carries the blood of the namesake of their race, the goddess Hylia." -pg. 93
"In ever age where Ganon rises up to cause chaos, there are born two defenders fated to protect the kingdom: a warrior with the soul of the hero and a sacred princess who is the goddess reborn". -pg. 366
"Up until that point the royal family's sacred power made them the authority in Hyrule, but because of the cycle of prosperity and decline of the kingdom of Hyrule, their technology was not able to keep up with the Sheikah's, creating a massive technological imbalance. In particular, the ability to create and control gigantic mechanical structures and wield an amount of power massive enough to subdue Calamity Ganon was said to be on par with the powers of the goddesses, and because of that, the people of Hyrule began to fear the Sheikah." -pg. 368
"The reincarnated goddess was said to have been an ancestor of the royal family of Hyrule..." -pg. 368
"... But share the common goal of eliminating the chosen hero and the inheritor of the blood of the goddess." -pg. 369
"The princesses of the royal family each possessed a sacred power... The queen herself possessed those sacred powers and was to tutor Zelda in its awakening and use..." -pg. 372
"Alternatively, it's possible that the fortune teller may have been the queen of Hyrule. Women of the royal family have the sacred power to seal but have also been known to have premonitions." - pg. 374
"Women born into Hyrule's royal family were said to be sacred princesses, inheritors of the blood of the goddess Hylia and the power that comes with it. Many princesses were naturally adept at awakening and utilizing this power. Princess Zelda's mother was said to have felt the power overflowing inside her." - pg. 378
"Unfortunately, the technique for unlocking the sealing power is not written down in any book; it is something that either comes naturally or develops through mentorship with one who has already awakened their power." -pg. 379
Phew, that's a lot. Mostly, we can extrapolate some very important points here. There is no mention in CaC of Hylians being biologically related to Hylia, only the the Royal Family. Also, to expand on what I said either that princesses are special, CaC makes it very clear that all the women of the Royal Family inherit sacred powers. Furthermore, more than one member can use it at a time, since Zelda's mother was supposed to each her the secrets of its power. CaC gives the impression that only the women are magical, but we know from previous games that other royal male members have been magical too. They simply lack that sealing power.
Considering only women have this sealing power, this could be two things: 1, magic, as I have previously discussed. Or 2, maybe the Y chromosome suppresses this power. (I don't know, I'm making guesses).
Realistically, this power would extend beyond the Royal Family over time, but since some royal members just... lack the innate ability to use it immediately, it may have been lost once you get beyond the royal family. If royal members only have one child at a time, then it would pretty much stay isolated. (We know that in AoL there are multiple children but perhaps by the time of Botw, policies changed, who knows?).
CaC also makes a point about the power of the Royal Family is what made them the authority of the land. But when it was challenged by Sheikah power said to rival it, people feared it. Even the king had them lock it away. This is in line with my prediction that the Royal Family may have purposely kept these powers isolated to their family, since being challenged was fearful. I'm not suggesting every king was power hungry, but lets not deny that every king wasn't perfect and good. Not even every queen.
The book only makes one mention of the princess being the goddess reborn, but considering that every woman in the Royal Family has this power, it's clearly not just Hylia's soul that is a requirement. It's her bloodline.
However... then you have Zelda Encyclopedia.:
"Hylians were the first race to establish civilization in ancient Hyrule. When the goddess Hylia took on a mortal form, she chose to live among Hylians." -pg. 14
"In the past, Hylians were able to wield magic of considerable might, but, over time, that ability has faded. However, because they are of Hylia's bloodline, the royal family, especially its princesses, is still strong in magic. Hylia's power has manifested whenever Hyrule is in danger." - pg. 14
"Eternally reborn, she (Hylia) lives among her people as a Hylian in the kingdom now known as Hyrule". - pg. 12
"When wars break out, those bearing the rightful lineage of Hylia are called to the kingdom of Hyrule to keep chaos at bay." -pg. 15
"She (the Zeldas) is the goddess Hylia reborn, and wield great power that she uses to keep evil in check," - pg. 16
ZE is contended in the fandom for it's... questionable accuracy and canon to the lore. But I still want to look at these quotes anyways. The book, firstly, calls humans "Hylians" no matter the era, including SS. Since Hylians do not exist in SS in name, this is kind of questionable itself. I also see no claim of Hylians having Hylia's bloodline, only the Royal Family. This is more in line with CaC, with HH being the only stand-out to claim otherwise. Realistically, HH is more accurate considering how genetics work but... Well, we don't get too deep into that, I've already discussed this.
The big difference here is ZE's claim that Hylia is eternally reborn, and that Hylia's powers manifest whenever Hyrule is in danger. Well, the second claim simply isn't true. We know that Hylia's power, from CaC, is just always present in the women of the Royal Family, no danger necessary. And there are two times in history that we don't really see Zelda's powers at play at all. In fact, a few more times than that, Zelda's power is either not stated or ever used. Zelda's power seems to have either been ineffective or not used in the Era Without a Hero, and in the Imprisoning War, it was only a piece of the puzzle (assuming her ancestor, the sage, was another Princess Zelda. This, however, is not stated anywhere, simply that the person was a Sage). LoZ Zelda, AoL Zelda, and OoX Zeldas never use their powers at all. You could even argue neither does MC Zelda's power- the power she uses is that of the Light Force, a separate power from Hylia's, and the magical wishing cap.
As for Hylia eternally being reborn.... That's hotly debated, I see. CaC only makes one reference to the princess being the goddess reborn, but one could argue the language is poetic in nature, not literal. This could extend to the language in ZE but I have a harder time seeing it that way. Every Zelda could be the Goddess reborn, in that she is reincarnated again and again. Or, she could be a metaphorical goddess reborn (except SS Zelda). We already know that Hylia's soul is not required to have the sealing magic per CaC. Furthermore, by the end of AoL, there are two living Zeldas. Either Hylia's soul splits (unlikely), not every Zelda is Hylia reborn (more likely), or Hylia was only reborn once and the rest are just her descendants (more likely).
I would like to argue that since Hylia is reborn into the royal family (which is her own descendants) again and again, this is why only Zelda has the sealing power and that's why it does not go beyond the Royal Family. But this logistically isn't likely with the text from CaC. That is not to say Hylia couldn't be reborn again and again. It's simply just fact that Hylia's powers manifest without her soul.
I know this was a very long post but I feel it's important to look at what the books say, as well as the games. Alas, the games say very little over the true nature of Zelda's powers overall, it's more in the actions than the words, and they vary on the princess. The books offer more insight, and are either older and lack newer game info (HH), or contended in the fandom and full of questionable information (ZE). CaC seems to be the most accurate information we have on the subject overall, since it's newer and comes from better sources.
There is one more concept someone else mentioned that I didn't bring up but should quickly touch on. Incest. Yes, I know, an uncomfortable topic. But the royal family in Japan for several eras practiced it to keep the line "pure". It's not a comfortable topic and I doubt Zelda would really touch on that (Zelda usually shies away from heavy topics and uncomfortable implications, only briefly mentioning something and never expanding upon it) because... it's kinda a children's game. Princess good, royal family good, that man over there bad. It usually stays simple.
I do think Nintendo made the mistake of adding Hylia's bloodline for this very reason, though, since it obviously makes no sense unless you just go with the "it's magic!" without trying to come up with multiple explanations like I did in this entire post. It was simpler when Zelda was just a princess and she occasionally had magic or something. But now, they have made their commitment and here we are, wondering angrily...wtf?
