We already know a timeline split was obvious, considering that The Wind Waker and Twilight Princess are both games that follow the Hero of Time's adventure, but from different endings.
Adulthood Timeline has Link sent back in time by Zelda after his battle and being absent from the timeline. Ganondorf breaks free of his sealing and Link is not there to stop it. So the people pray and the Gods they praised flooded their world in order to keep Ganondorf from obtaining the pieces of the Triforce he sought.
Childhood Timeline picks up where Link was sent back in time by Zelda to a point before he drew the Master Sword but after his first meeting with Princess Zelda. Link had accidentally brought the Triforce of Courage back in time with him and, as was stated in Ocarina of Time, when one person possesses a piece of the Triforce, the other two pieces autoatically are attracted to the individuals that best represent the trait their piece represents.
Link goes and has his adventure in Termina while Ganondorf is sentenced to death... only to have his execution thwarted by the Triforce of Power and he is cast into the Twilight Realm, directly leading into Twilight Princess's earliest events.
As for why the Gate of Time and Harp of Ages do not have the same rules when you go back in time to alter the past?
We have only seen timeline splits happen with the Gate of Time/Door of Time.
Skyward Sword has a timeline split and we follow the timeline where we planted the Lifefruit Tree but disregard the timeline Link accidentally abandoned. Groose's alteration of dialogue is evident of this as before you go back in time he comments about the empty grove in the Sealed Temple. But after you return to the present to harvest the Lifefruit, Groose's dialogue directly states that the tree has been here since he followed you from Skyloft.
Ocarina of Time has the Childhood and Adulthood timelines split off as a result of the time shenanigans that were performed during the game. As for the Downfall Timeline, it doesn't make sense to have it be a result of Link dying during the final battle with Ganondorf/Ganon because there was no time-travel involved in the last fight. However, it DOES make sense if it is simply a timeline Link accidentally abandoned when trying to complete the Sprit Temple, as part of the temple had to have the player go back in time to alter the past and solve half the dungeon. Returning to the future, you complete the other half. Ergo, the Downfall Timeline SHOULD have a Spirit Temple that was never completed.
Meanwhile, there is the Harp of Ages from Oracle of Ages.
Even though Veran traveled back into the past and altered history while Link and Impa were still in the "present era", both Link and Impa were not altered by its time-travel effects.
If the Harp of Ages made timeline splits, then that would only mean that at the start of the game, Veran possesses Nayru the Oracle of Ages, declares her sinister intentions, jumps back in time... ... and nothing happens. Because if the Harp of Ages followed the same rules as the Gate of Time/Door of Time, there would be a timeline now where Veran and Nayru just vanished and ceased to exist and there would be two more timelines made -
The Tower Timeline in which Veran was successful in lighting her flame for the ritual.
The Normal Timeline in which nothing happened so the status quo never changed. Just that this timeline now would no longer have the Harp of Age and an Oracle of Ages.
Instead we see the direct results of Veran's influence in the past as it alters the very present we are standing in, thus we play through the entire game to correct all the wrongdoings of Veran and restore the "present era" into being what we knew it to be before her negative influence on history began.
Gate of Time/Door of Time causes splits of the timeline.
Each split timeline continues onward into the future, with your departures into the past resulting in that timeline suddenly being absent of you and time marches on as normal.
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Harp of Ages causes you to venture into the past to alter a single timeline's events.
Basically it is like playing a game with multiple choices, you want to have the best choice for one playthrough.
You make a bad choice? You load a previous save and then overwrite the save that had the bad choices.
Repeat until you achieve the best possible ending in the game.
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As for the Door of Time sealing Link?
It had to have sealed Link.
Rauru stated that he was in the Sacred Realm for seven years.
Ganondorf entered the Sacred Realm.
Ganondorf is a smart tactician.
Wouldn't it be a smart move to keep an eye on the Hero and take him out before he can meddle in your plans?
Well, if Link is in the Sacred Realm for seven years and under some sort of protection, then it makes sense why Ganondorf wouldn't do such a thing. He couldn't so much as scratch Link. So, he had to scheme and work around that. Instead, he turned Link into a pawn that would unintentionally help him reveal the location of Zelda and the other Triforce pieces, Ganondorf not truly knowing until the endgame that Link possessed Farore's piece.
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Not that hard to do actually.
All they have to do is make a few games with odd details mixed into the ends that make little to no sense UNTIL we get the game that explains the timeline merger. It would be an interesting concept.
Imagine it, a BotW style 3D Zelda game in which the Downfall, Childhood, and Adulthood timelines are set to converge by mortal hands seeking to meddle with time.
A failed initial attempt by these villains results in a landscape covered in temporal rifts, pockets in which "islands" of the other timelines are scattered across the world. You step into these island and your modeling and lighting shift. One timeline is in Realistic. Another in Toon styles. And the third in Skyward Sword painted style.
Three Kingdoms of Hyrule with three parallel Bloodlines of Hylia.
Three Demon Armies led by Three separate Villains.
Three Links, two of which being the ones we played as in two other games with those "peculiar endings" are now NPCs that we can encounter as we play as the third Link that is to be the one who helps mend the timeline.
Merging the timeline introduces the BotW semi-realistic cell-shaded art style; artistically signifiying that the three timelines are now one and the same. The game eventually has its happy ending and the timelines are reunified, allowing the Breath of the Wild games to be placed afterwards comfortably.
Skyward Sword is the definitive first game in the original Unified Timeline.
The Ancient Tapestry of 10,000 Years, Breath of the Wild, and Tears of the Kingdom take place on a second Reunified Timeline at the end of the timeline chart, with other games possibly happening before the Ancient Tapestry events and after the BotW/TotK.
And possibly even other games on the Terrako Paradox timeline caused by Hyrule Warriors Age of Calamity.
If Nintendo were to add in a set of games that does this and reunites the split timelines, it would allow them to add the BotW games to the timeline definitively and, more importantly, comfortably. Especially considering that currently they are not concerned with it but it is something that they know might come up in the future. For now, they want us to be imaginative with it. Though BotW takes place at the end of the timeline, they currently do not want to focus on specifically where. Later on, if they do want to place it definitively; the best solution is to have some games eventually reunify the timeline and THEN place BotW/TotK there.