Zelda is the title character for a reason.

Here's the in-game opening story description for the original NES Legend of Zelda.

LONG AGO, GANON, PRINCE OF DARKNESS, STOLE THE TRIFORCE OF POWER. PRINCESS ZELDA OF HYRULE BROKE THE TRIFORCE OF WISDOM INTO EIGHT PIECES AND HID THEM FROM GANON BEFORE SHE WAS KIDNAPPED BY GANON'S MINIONS. LINK, YOU MUST FIND THE PIECES AND SAVE ZELDA.

And here's the more detailed version in the Instruction Manuel.

A long, long time ago the World was in an age of Chaos. In the middle of this chaos, in a little kingdom in the land of Hyrule, a legend was being handed down from generation to generation, the legend of the 'Triforce'; golden triangles possessing mystical powers.

One day, an evil army attacked this peaceful little kingdom and stole the Triforce of Power. This army was led by Gannon, the powerful Prince of Darkness who sought to plunge the World into fear and darkness under his rule. Fearing his wicked rule, Zelda, the princess of this kingdom, split up the Triforce of Wisdom into eight fragments and hid them throughout the realm to save the last remaining Triforce from the clutches of the evil Ganon. At the same time, she commanded her most trustworthy nursemaid, Impa, to secretly escape into the land and go find a man with enough courage to destroy the evil Gannon. Upon hearing this, Ganon grew angry, imprisoned the princess, and sent out a party in search of Impa.

Braving forests and mountains, Impa fled for her life from her pursuers. As she reached the very limit of her energy she found herself surrounded by Gannon’s evil henchmen. Cornered! What could she do? … But wait!

All was not lost. A young lad appeared. He skillfully drove off Ganon’s henchmen, and saved Impa from a fate worse than death. His name was Link. During his travels he had come across Impa and Ganon’s henchmen. Impa told Link the whole story of the princess Zelda and the evil Ganon. Burning with a sense of justice, Link resolved to save Zelda, but Ganon was a powerful opponent. He held the Triforce of Power. And so, in order to fight off Ganon, Link had to bring the scattered eight fragments of the Triforce of Wisdom together to rebuild the mystical Triangle.

If he couldn't do this, there would be no chance Link could fight his way into Death Mountain where Ganon lived. Can Link really destroy Ganon and save the Princess Zelda? Only your skill can answer that question. Good luck. Use the Triforce wisely.

What you'll notice is that while the player plays Link you are on a mission from Zelda, you are attempting to carry out her plan.  This is why even from the very first game in 1985 she had agency that Princess Peach Toadstool did not.  Now today Peach has become a character with as much agency as Zelda and that's great, I don't wanna a start a Catfight between Zelda Simps and Peach Simps.

In Adventure of Link this aspect of Zelda's role is gone but it's back in A Link to The Past where while you can read a backstory about both the distant and recent past, actual gameplay begins with the Player as Link responding to Zelda's call to adventure.

However in Ocarina of Time Zelda isn't who sends link on his quest and when you do get a mission from her it winds up being the McGuffin Delivery Trope which I hate when it's done for Video Games, I don't want to be told things would have been better if I did nothing.

I like every Zelda game ultimately, but the reason why I consider OoT overrated in the story department is because I care little about how much more cinematic things are like now in 3D if it lessens one of my favorites aspects of the classic games.

Skyward Sword's extreme linearity may be the opposite of the original NES Game in how you play but it was the first game (at least in the main series) since ALttP to bring back this aspect of it, Link is carrying out Zelda's plan, well you could get pedantic and say it's Hylia's plan but they are ultimately the same Entity, and Zelda proper is playing her part in getting the ball rolling.

And then Breath of The Wild is also very much back to the original premise.