this has has been sitting in our inbox for a long time bc i’ve been trying to think of the best way to answer it so here we go~

the most obvious answer is that there would not have been a story - or, at least, the same story. perhaps someone else would’ve come along some many years later and discovered aang and appa, but would aang still have been able to accomplish all he needed to do without the help of katara, sokka, and eventually toph, suki, zuko, etc.? i don’t know man

zuko was pretty determined to find the avatar. it was his quest - his obsession, even - for three years, and didn’t ever really show any signs of backing down despite the slim odds. because of this, one could likely conclude he would have continued this way for a long, long time. however, at the heart of everything, zuko was still a kid. he was 15 or 16, right? he’d already been through so much even before being sent on what was almost universally considered a wild goose chase. regardless of how determined you are or what you have at stake, all of that gets to you eventually. maybe he wouldn’t have given up, per se, but it would be eating away at him more than he let on. that said, i’d also like to believe that somewhere along the line, iroh would’ve successfully helped him in breaking away from his obsession, but there’s really no telling what would’ve had to happen to the both of them in order for that to happen, which is another reason why it was so important that zuko eventually did realize he was fighting for the wrong side of the war.

and about the korra part - i haven’t watched korra yet, so it’s a little harder to say. the state of the world would most likely be drastically different, and by that point it would probably be up to someone else to find the avatar - and the result of that all depends on who. i think aang’s guilt would’ve been much, much heavier, but he would still try to fulfill his destiny as the avatar.