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Listen. There are many funny ideas of where Zuko just stumbles through the world and just finds the gaang? How he finds the avatar that has been missing for 100 years? And manages in the vicinity when he breaks free?
Even in the beginning it sounds like Zuko has been all around the world at least multiple times. While I want to talk about the people and places he met while during the three years before the beginning of canon, this is about how Zuko finds the gaang when he just stumbles upon them.
At first, I wanted it to be some spirit interference like in Embers fanfic, that the spirits need Aang to do avatar things, he slept long enough and without motivation Aang would just stay in places and be an air nomad and kid to just play and fly away. But with Zuko he is in a hurry to do so - but still manages to get to places and people to do avatar things at a faster pace than if Zuko wasn't chasing them. The spirits just guiding him to the avatar.
Another was that during the three years before canon, Zuko chases spirits and their mentions because the avatar is the bridge between spirits and humans, CLEARLY there is some relation and the spirits might know. So he bothered spirits and enough of that happened (he and his crew manage to get out and alive through all these encounters) that they are like, 'listen. avatar. wake up, this guy keeps bothering us about your location. so get up, be avatar and please stop this guy bothering us so he can bother you'. And if he isn't in the avatar's vicinity - well he might come back so here's some hints/be near him to stop coming to us.
Then I clearly thought of something brilliant. Zuko is related to Roku. The PREVIOUS avatar. Like, the literal previous avatar before aang. So there must be a connection. Not just a blood connection but some connection spiritually given that this is also the avatar. So he has an inner compass or something to that pings 'relative! avatar! avatar relative!' and just is a blood hound to get near his location and then a lucky break happens to get him more clues.