Aang already had the ability to go into the avatar state even before mastering all the elements. While I wouldn't say he "mastered" elements per se, he would learn bending a lot quicker than Korra ever did. Hell, he basically became a fire bending masters in ONE episode "The Firebending Masters'.
So I don't really understand why people rag on Korra so much for getting everything so easy. Especially when the plot takes place in her late teens (where she already had training). Although I do agree there is some conflict in continuity. For instance, most babies don't know they are the avatar. And they aren't told till they are 16. The fact that Korra could bend three elements as a kid was goofy, and feels kind out of place with Avatar.
Even the ending where Aang gives her all her powers back, how many times in Avatar did the Avatar State basically act as a cure all for Aang? It happens quite often in Avatar.
EDIT: I do agree with those saying she didn't "learn" what was actually important though. However, isn't that apart of her character? Aang was a monk, so it makes sense he was more willing to be in touch with what really mattered. He was much more introspective. That might explain why he could go into the Avatar state even before mastering it, and Korra could never do so (because she didn't know how to get in touch with her spiritual side). But you guys are right about the fact that, she spent all that time training - and yet her personality traits don't really line up with someone that went through all that training. For instance, her suggesting that they cheat at the pro-bending match seems out of line with an Avatar.
Not really, since when it comes to how Avatars behave, we only know what Aang would do. We can reasonably believe what Roku would've done, but less so than Aang. We barely know anything about Kyoshi, Kuruk WOULD'VE cheated, you can bet, and Avatar Susan...ahem, Yangchen, probably would've not even bothered with pro-bending haha. Avatar Wan's story is a very exciting thing to look forward to in Book 2 precisely because we will see how the Avatar Spirit choses who can be the Avatar or who can't, if it choses at all, and if it can get it wrong.
But, I think we do depend too much on Aang for what we expect from an Avatar, and I think that Korra is Bryan and Mike's way of saying that most definitely not all Avatars are like Aang, or that the only way to be a great Avatar is to be an Airbender one.