Two of things that get thrown out about Kvothe is he is a mary sue, and an unreliable narrator. Both are not true because the subtext of the stories is hinting at you the whole time that they are not. Kvothe KNOWS his weakness now, as Kote the inn keeper, and if you know it too, you realize he is not unreliable in the sense what he says is wrong or he made it up. Everything he says is correct, as he saw it at the time.
It’s like you telling a story about yourself, before realizing your own ignorance and blind spots, and then the former story becoming ironic when you reveal a misconception or many you previously had. That’s the set up, you are meant to read the third book (or you were meant to read it) and cringe at how Kvothe acted in the first two, how he was mistaken, because he thought he was wiser than he was. If he is just lying all the time in the first two, nothing means anything and everything can mean anything, in the dark all the cats are grey
Secondly, neither Kvothe nor Kote are mary-sues and the previous point supports this one, Kvothe has a fatal flaw like greek hero’s, he actually has a couple. He is impatient, he readily disregards rules, and he ASSUMES too much. There is nothing more wrong than an intelligent person who is wrong, because they can use their intellect to twist every variable to fit their ends. That’s Kvothe. His worst flaw is his impatience.
If Kvothe was a mary sue we would be reading something more like Eragon