In short, yes. She does matter in the overarching theme of the Legend of Zelda. She is the protector of the Triforce and the Hylian people, battling against the wicked forces of the demons who intend to use it for malevolent purpose. She is the one who chose Link as the hero and made go through the her trials to prepare him for battle. She is the reason why Zelda and the Royal Family is important to the whole legend.
When looking back at lore and legends of each Zelda game, there's an overarching theme: The Legend of the Hero, the Princess, and the Demon. From the beginning, there was no really mythos or lore about the world of Hyrule. We've got bits here and there in the original Legend of Zelda and the Adventure of Link, but nothing really spectacular. The Triforce is a magical object that can grant wishes, Ganon is a generic evil demon that wants to rule the world, Zelda is the classical princess, and Link is your everyday hero. But the simple fairy tale of the NES era evolved into something deeper and more legendary in A Link to the Past. The Triforce is now an artificial of the gods, Ganon was once a greedy thief who became a demon pig when he touched the Triforce, Zelda is a descendant from the seven sages and Link is a descendant of the knights who battled Ganon's forces. The theme is developed further in Ocarina of Time, by having Link, Zelda, and Ganondorf inherit a piece of the Triforce, which will determine the fate of the world.
But somewhere along that time, Princess Zelda started to lose her relevance among the Triforce trio. It's now Link the Hero against the Ganon the Demon, the classic good vs evil. But where does Zelda fits in to all of this? What's her role, mentor to Link? If so, then she is easily replaceable with characters like Rauru or the King of Red Lions. Special powers? She does have magical abilities such as calling the spirits of the light or transforming into Shiek, but it's rarely emphasized on and often declared to be the result of the Triforce of Wisdom. And why is that important?
When adding Hylia and the Ancient Battle to the mix of the Triforce trio, Zelda's role is suddenly more prominent. As the incarnation (or descendant) of Hylia, Princess Zelda is essentially the protector of Hyrule and the Triforce that lay within. As such, she is usually the one who would prompt Link on a quest to save Hyrule, be it separating the Triforce of Wisdom into eight pieces or sending him to find the Master Sword. She is kind and forgiving, even to those that have shown no kindness to her, and would sacrifice herself to save her people and those she loves. And as the vessel of Hylia's powers and the protector of Hyrule and the Triforce, she is often a target for villains wanting to rule the world.