Also, I think he's more of a deconstruction of a Gary Stu/Mary Sue than an actual one, given that -
1) he's the one telling the story, so his achievements are aggrandized and his faults are swept under the rug ("I'm not good at alchemy, it's weird, and not important") and makes excuses for himself ("I couldn't answer the question, it must have been Hemme's fault" "I was dumb enough to bring an open flame into an ancient parchment-based library, but I was tricked so it's not my fault")
2) Elodin basically outclassed him at everything. One of the hallmarks of a mary sue is that they are the best at everything.
3) He's monsterously flawed - egotistical, overconfident, impatient, and obviously all these things lead to some sort of tremendous downfall which is how he ends up running a backwater inn under an assumed name instead of being the Greatest Hero Of All. Which also brings up...
4) Another crucial hallmarks of a mary sue is that by the end of the story, everyone loves them. Kvothe easily makes more enemies than friends at every turn, and most of them are not Bad Guys. By the end/beginning of the story, he's so hated that he's got a price on his head and has to live in carefully guarded anonymity in east bumfuck village.