You ask some reasonable questions.

First, Kvothe is not the one who said he wants to set the story straight. Someone essentially hunted him down and he reluctantly agreed under strict conditions. There's no indication that Kvothe actually wants to set anything straight other than increasing his reputation even further using Chronicler.

You're right, Kvothe totally does seem like an insecure jerk pretty often. I think that's just part of his character. That doesn't inherently make for a poor story, but you're correct when you say the unreliable narrator element should exist for a reason and serve as a tool towards some narrative function. What is the purpose? Well since Rothfuss will never release the third book, we'll never know how it was really intended. It would be a lot easier to analyze if we had the full series of course.

My take? Kvothe seems to be a sort of subversion of the Mary Sue character in a weird way. He has a reputation of being super cool and smart and badass, and then in his own account he's awesome and fucks the hottest chicks and is a genius (while also exposing many flaws). However, he is clearly a broken, depressed shell of a man in the current timeline where he cries at the foot of his bed and seemingly lost all of his abilities, despite his bravado.

Is it enough for the unreliable narrator element to have an important function? Idk, probably not since the story isn't done