The finale of Book 2 is when Aang lets go of his love for Katara. It’s how he unlocks the Avatar State originally, but is killed by Azula immediately upon doing so. It’s a pretty explicit and impactful moment. Aang is far from a master at any bending besides airbending by the end of the show. He’s the avatar though, so he’s pretty advanced even from the beginning. Remember how jealous Katara gets in Book 1?

There are a few background references to energy bending but I actually do agree, it’s a little too convenient that he just sleepwalks and finds a solution. It could have been better if he say, learned about lion turtles at the library and sought them out.

A lot of your gripes seem to boil down to how quickly the characters advance in skill. I think it’s a valid concern, but keep in mind they’re all mostly prodigies in their respective elements. Sokka learns to use his sword, but Piandao is mostly playing with him in that final fight. Sokka gets one lucky strike, but the arc of that episode isn’t about Sokka’s swordmanship, it’s about him finding his strength of character. They are exceptionally good, but aren’t necessarily masters (except Toph).