What TTRPG(s?) do you think might be good for doing Avatar: the Last Airbender/Legend of Korra? I'd especially like one that makes stating out benders with the unusual powers (earth benders who can do metal or lavabending, fire benders who can do blue flame or lightningbending, etc.) possible and mechanically distinct from the basic forms. Any level of crunchy is fine as long as it's not too hard to match the tone of the shows. Thanks!
I’m assuming by “mechanically distinct” you mean you don’t want style differences to boil down to “your +1 bonus is a different colour”, which rules out most of the rules-light options - so really, it’s a good thing you’re okay with any level of crunchy!
That said, it really depends on what you’re looking for gameplay-wise. If you’re keen on big set-piece battles, Dungeons & Dragons 4th Edition honestly isn’t a bad way to go - it’s basically “D&D does martial arts action movies” anyway, and with a bit of judicious reskinning of class powers you can end up with an enormous variety of elemental martial arts that work very differently in play. Plus, the game balance is tight enough that you don’t have to worry about non-magic-using PCs being made irrelevant in battle by their fireball-chucking peers, which is a. important for your Sokka/Ty Lee/Asami-type PCs, and b. something that very few other games will give you if you’re playing by the book.
If you’re after tactical set-pieces, but you want something that handles them a little more abstractly than D&D4E, you could also take a look at Legends of the Wulin. I’ll level with you: it wouldn’t be my first choice: the core book’s organisation is nearly impenetrable, and characters who don’t use supernatural martial arts are basically irrelevant save as cannon fodder. However, if you can work around both of those shortcomings, its handling of martial arts - both during character creation and in play - is a frankly fascinating piece of game design that very readily lends itself to Avatar’s brand of high fantasy weirdness.
(The other major caveat is that LotW’s current publishers are a pack of crooks who’ve straight-up defrauded several game designers of my acquaintance, so whether you want to put money in their pocket… well, it’s up to you.)
On the other hand, if you’re not keen on lengthy tactical play, that’s a taller order, since there’s far less design space for keeping different styles of magical punching mechanically distinct. It’s not impossible, though. Feng Shui 2 could be a good starting point, though you’d need to homebrew most of your own archetypes. Anima Prime is another way to go, if you don’t mind having to write up your own skill system (the game’s default one is rather too dieslpunk-centric for Avatar) - plus it’s free, which is always fun.