Take Toph for instance: She's IMMEDIATELY introduced as an overpowered god of her magic system. There's no question about it. And yes we learn that it's because she is a nearly self taught guru, but her skill is then very rarely seen as the solution to an episode.

I think there is more to it than that. If it had ended with her being a completely OP Earthbender, it would have made her a lesser character. Instead, we are shown the limits of her powers almost immediately after she is introduced (in the form of her being useless the second something stops touching the ground). Which I think helped to improve her character even from the beginning in a way that a lot of modern movies no longer do (see: Rey besting Kylo Ren, without any followup to show how that happened).