We have to keep the author's intentions in mind. A Mary Sue is everything the author wishes himself to be. If the author wishes he were a super-awesome arrogant prick, that's a Mary Sue. You may have a problem with that, but that's irrelevant.
By this logic, if the author wishes to be a fully realized character with plenty of flaws and limitations, interacting realistically with everyone in a cast and by all means a normal person within the universe, then it's still a mary sue.
And I don't think we have a right to think that Kvothe is EVERYTHING Pat Rothfuss wishes to be, like a murderer.
Again, like my point above, your view on what's right is irrelevant to a Mary Sue. It has to be viewed in-universe as presented by the story. Is his arrogance portrayed negatively? Maybe you could argue for sometimes, but overwhelmingly he's shown to be right, and his problems are from others not understanding or not keeping up.
His arrogance is ALWAYS portrayed as wrong, even when he's right. Being right about something doesn't give you the privileged of acting like a wrongly about it. Just look at how Kilvin is constantly on his ass about adhering to morals in making artifacts even if he finds the artifacts brilliant. The morals come first, a lecture about how wrong he was to do something, a punishment for it, THEN he accepts the artifact. Nobody likes him when he's "Aw, yeah, check out how awesome I am", which is why he mostly keeps it to himself.
A Mary Sue is EVERYTHING an author wants to be, in a given setting. It's not a Mary Sue if it's just a part of the author. It's the totality of it that's the problem, and how this warps all other elements of the story: no inner struggle to overcome, no character development other than becoming more awesome, no other characters with depth outside of their relationship with the protagonist, etc.
Again, this idea comes from fan fics because it's simple, bad writing. "My guy's going to be, like, the most strongest, and he'll beat everyone up, and everyone will love him."
If that's the definition your going with, then Kvothe is
definitely not a mary sue.
Inner struggle? The best one I can think of offhand is how he is horrified by how own murderous urges as when he tortured the false Edema Ruh. There are others scattered throughout the book.
Kvothe changes over the course of the story, in subtle ways like how he finds his strive to be a hero, and other more obvious ways, like how his older self resents his youthful idiocy.
And Simmon has a good depth to him for how little he is in the story. He is a genuinely good person, but he hails from a very politically charged family, where he is somewhat of a black sheep for wanting to study things like poetry. Then his relationship with woman is to wear a mask of joviality, but people just consider him an idiot for it and disregard him, leading him to have major problems self esteem problems. When people take the time to know him and his most serious side and rather dark place of origin, where they used to torture people for medical information, not unlike the Nazi Josef Mengele, it's kind of a miracle that he came out as such a genuinely altruistic person. Perhaps that's why he came out that way. Elodin is good too, with his secret past. Also the mysterious storyteller in the first book the one that took care of him in Tarbean.
You can argue it all you want, but he personally said he wishes he were a polymath. And people have been complaining that Kvothe is a polymath since way before Rothfuss said that. And you have to adjust skills for setting and story: magic duels for martial arts where appropriate, and music playing for writing.
You just said that the mark of a mary sue is that he has to want to be Kvothe in his totality. This would have to include having his parents horrifically slaughtered (Note that this is a guy whose book came out right after his mother had died from cancer and he was so shaken up about it that he cries even now) murdering, being the start of a terrible war that is getting a ton of people killed, and several other unfortunate things. For Kvothe to be a mary sue by your logic, he's have to want the his parents dead and the blood of hundreds on his hands.
But don't you think the frame story is more interesting? The Kvothe there at least has an inner struggle to get over: he doesn't believe in himself. I'm way more interested to find out what happens in the frame story than the main story. Yeah yeah, so young Kvothe will go kick some more ass somewhere else, whatever; but how is adult Kvothe going to get over his self-doubt? Will he learn to enjoy life again? That's interesting to me.
What is interesting doesn't matter, either to you or me. It's irrelevant to the discussion. But you are factually wrong to say that Kvothe doesn't have inner struggles or is just there being effortlessly awesome with no problems or flaws to speak of. Really, it's to the point where I want to suggest you go back, reread the book and focus on why kvothe's problems exist in the first place, or how the Elodin regards him as an idiot not because he's an eccentric old man but because kvothe actually does stupid things in front of him, or how he wouldn't solve a good portion of atleast 2 major problems without Wilhelm and Simmon exhausting themselves covering his ass with kvothe fully acknowledging how they are much better friends than he deserves and owes them so much.
Tyrion Lannister would be a Mary Sue if GRRM didn't make him an amoral dwarf, as well as spending a lot of time contrasting him with Jaime.
He was very aware of the pitfalls of self inserts so he set gave Tyrion a pretty severe flaw, and the character is much better for it. Kvothe, in the context of Kingkiller Chronicles, is like if Tyrion had the body of Jaime.
It doesn't matter. By the wishfulfillment logic, if GRRM wants to be as smart as tyrion, then despite his flaws, he is a mary sue because it's wishfulfillment. Hell, even the amorality can be seen as a virtue, if your subscribe that GRRM wishes he could be a dick without repercussions. But if having flaws means that a character isn't a mary sue despite wishfulfillment, then just read through my posts here outlining Kvothe's numerous flaws because I'm tired of retyping them over and over for people who ignore them.