
Stephanie C. answered 06/19/19
English BA, copy editor and proofreader can improve your writing!
In most cases, it's both a case of inherent potential and an active choice to fulfill that potential which makes a hero. No matter what powers you have, it doesn't matter if you never use them. If your character is a god-like figure but sits out every conflict, the impact on the world of your story is the same as if they had no powers at all. The higher powers might as well not exist.
Some of our earliest "chosen" hero figures come from Greek myth. The great heroes are often demigods, so they have extraordinary potential and usually a lot of outside assistance, but they have to make the choice to step out of relative safety or obscurity (often unaware of their divine parentage) to take on a quest of some sort. The trials they endure in the quest test their worthiness. When they are clever and brave they are rewarded. When they get too full of themselves, though, they are punished for their hubris.
Achilles is a great example of choosing to be a hero. He was the son of a mortal king and the sea goddess Thetis. He was prophesied to live either a long life but be forgotten in due time, or a brief and glorious life during which he would die young but be remembered forever.
His mother wanted to keep him out of the brewing war between the Greeks and the Trojans, so she hid him in the court of a foreign king, but Odysseus discovered him and convinced him to join the Greeks in battle. So, he could have been a demigod with all the potential to be a "chosen one", but if he had remained hidden or just wanted a longer life, he never would have become a hero by fulfilling his fate. Both halves of the story are necessary.
Are you familiar with Joseph Campbell's work? His concept of the Hero's Journey is a good framework by which to understand the phases of a character who is plucked from obscurity to become some type of chosen hero figure throughout a quest story. There are points along the way at which the character has to make choices. Choose one way and you become a hero. Choose the other way and maybe you end up as a villain. Free will is always an important element. (Refuse to make any active choice and I guess there is no story at all, just entropy).
As for your paradox, I think it's more like two sides of the same coin. The passive and active parts of the character are both necessary at various points. Passive abilities like inborn higher powers or unique nature don't make a "chosen one" unless that person actively puts them to good use.
Or from the other perspective, just being the kind of person who chooses to take on a difficult and dangerous task, often for the benefit of others, is special in itself. Those ordinary heroes who don't have any obvious special powers effectively choose themselves. That initiative confers heroic qualities throughout their journey.