In the finale, everyone ends up fulfilling their destinies. But three people in particular came to their destinies in a very similar way: they already had an end goal in mind but were able to find a different path to get there than they initially thought. You may only think of Zuko when hearing this, but it very much applies to Aang and Iroh as well.
Let’s start with Iroh. In “Sozin’s Comet Part 2,” he states, “when I was a boy, I had a vision that I would one day take Ba Sing Se. Only now do I see that my destiny is to take it back from the Fire Nation so the Earth Kingdom can be free again.” So, Iroh knew that his destiny was to take Ba Sing Se, but he thought it would be in the name of the Fire Nation; instead, he accomplishes this goal in the exact opposite way, by reclaiming it in the name of the Earth Kingdom:
So that’s Iroh. Aang’s story unfolds in a way more reminiscent of Zuko’s. Starting in “Sozin’s Comet Part 1,” we see Aang’s first hesitations at the thought of having to kill the Fire Lord:
I put a picture of Zuko from “Nightmares and Daydreams” on the right because what they say in their respective scenes is similar. Aang says, “it just didn't feel right. I didn't feel like myself,” after not being able to kill the Melon Lord in training, and Zuko says, “during the meeting, I was the perfect prince. The son my father wanted. But I wasn't me.” Both characters are thus going through a realization that this isn’t what their destiny should be.
Zuko himself even starts to see how similar his own story is to Aang’s. Later in the episode, Katara and Aang start arguing over having to kill the Fire Lord, and Aang storms off in frustration. Katara tries to follow him, saying, “Aang, don’t walk away from this,” but Zuko stops her:
He puts his hand on her shoulder and tells her, “let him go. He needs time to sort it out by himself.” Zuko says this because this was the exact same situation he went through in the first half of Book 2: he got extremely angry at his circumstances and ended up leaving his uncle to sort through things by himself, and by the end, he had gotten a much better hold of his identity (these events occurred in “Avatar Day” through “The Chase”). Therefore, Zuko knows best that alone time is the way that Aang will be able to figure out what to do, and this is why he stops Katara.
Continuing into part 2 of the finale, we come upon Aang trying to seek advice from his past lives on what to do:
However, we see that every last one of them hints at the same thing: he needs to prioritize killing Ozai over his morality. It seems that everybody is telling him he needs to follow a certain path, just as everybody in the Fire Nation (minus Iroh) seemed to be telling Zuko that he needed to follow a certain path.
In part 3, Aang demonstrates more than once that he’s still sticking by his pacifist morals. First, he tries to reason with Ozai before engaging in battle with him:
This continues Aang’s tendency of never resorting to violence unless absolutely necessary - even though he fully knows there’s zero chance of Ozai backing down, he tries to avoid fighting anyway because that pacifist aspect of him is so deeply ingrained in his values.
And next, when Ozai shoots lightning at Aang, Aang is able to use the lightning redirection technique he learned from Zuko, but when the moment comes, he makes a conscious decision to aim the redirected lightning away from Ozai as not to kill him:
This again just shows that Aang won’t sacrifice his values; he won’t change his morals just because others are telling him to. He has the inner strength to follow his own moral compass and not let others convince him otherwise.
Then, finally, in part 4, he proves everyone else wrong when he demonstrates that he was right to not sacrifice his own morals:
By using energybending to take away Ozai’s bending, Aang successfully creates his own destiny: he finds an alternate path - one that he follows because *he* feels it’s right, not one just succumbing to one that everyone else wants - to accomplish his end goal of defeating the Fire Lord and bringing balance to the world. Thus, Aang becomes the glorious fully-realized Avatar he was destined to be, but he does so by following his own heart.
Aang, Iroh, and Zuko, while all sharing the same overall story, are able to find their destinies in slightly different fashions. Aang finds his through pure determination to keep his morals intact; Iroh finds his by transforming a tragic event (the death of his nephew in the war) into a motivation for positive change; Zuko finds his through making mistakes and initially choosing the wrong path. So, when we think of the idea of finding one’s destiny in ATLA, we should realize Zuko isn’t the only character we can look to for wisdom. Aang and Iroh also have amazing stories of forging their own destinies, and they deserve attention in this regard as well.
To sum everything up, lemme just leave it at this line Iroh mentioned to Zuko in a flashback in “The Western Air Temple”:
“Destiny is a funny thing. You never know how things are going to work out. But if you keep an open mind, and an open heart, I promise you will find your own destiny someday.”