I'm not sure if this is canon or fanon, but I remember reading somewhere that the Fire Nation knew that the next Avatar would be born into the Southern Water Tribe. I think it had something to do with the previous WT Avatar having been from the Northern tribe. Anyway, I think this is why the Fire Nation focused on capturing waterbenders from the south. For all they knew, Aang was dead and the Avatar had reincarnated, so best case scenario, the Avatar would be locked up for decades without even knowing who he or she was, and worst case, the Avatar would need to travel to the opposite side of the world for training, to a location they know about, across seas that they control.
You know, that could be an interesting start for a previous generation fic, with or without Aang, so long as air travel isn't an option.