One of the Kingkiller books ostensible strengths is the way they play around with the conventions of story-telling in desconstructive and meta ways. In practice, this amounts to two main themes that run through the novels: the creation of Kvothe's legend, and revealing the more mundane reality behind the events that gave birth to said legend. Today, we're going to mostly look at the former.
If we were to consider a figure like Kvothe in the real world, we'd expect to find that most of their mythologizing was ad hoc, attaining legendary status first due to some incident or deed, and then people went back to examine their earlier life and slotted the stories they found there into the legend, twisting and inflating as needed. And some of that has happened with Kvothe, as his narration includes many incidents which Chronicler counts among the keystones of his legend, but which clearly passed by without notice at the time.
But Kvothe's reputation also starts growing from a very young age, more or less as soon as he sets foot in the University. In the last post, I talked about how he was sentenced to a whipping for injuring one of his teachers with magic; prior to said whipping, he takes a drug that slows his heart-rate in order to avoid bleeding. This earns him the first of his many nicknames, "Kvothe the Bloodless", as the students who witness the whipping immediately start spreading wild rumours about how Kvothe is some sort of supernatural entity.
Keep in mind, these are students at a prestigious institute of higher learning; many of them are probably studying botany or human physiology, or both, and would surely be aware of how Kvothe pulled the stunt off (he was able to find the means to do so just by asking another student, after all). But even if that isn't the case, these people are all studying various branches of this universe's version of science; surely they'd be more skeptical than this?
This pattern holds true for most of the incidents that made Kvothe famous: he does something strange or implausible, and everyone in the vicinity immediately jumps to the most outlandish possible explanation, then runs off to start recounting that version of events to anyone who'll listen. The only way this makes sense is if they've read the premise of the books they're in, and know that Kvothe will become famous.
Later, in the second book, Kvothe pops into a cafe and discovers that literally everyone inside is telling stories about him--and this isn't even the height of his fame. At this point, he has yet to do whatever it is he does to truly propel himself into the annals of history; his reputation is built solely on a string of implausible feats, none of which seem like they'd make much of a splash beyond the immediate area they took place in. Do I need to explain how unbelievable and shallow this is?
You also have to keep in mind the timescale that all of this is taking place over. Kvothe is fifteen at the time of his whipping, which is the first major step in his journey to fame; by the time the framing story begins he's roughly twenty five to twenty nine, and has been in exile for a few years. That means you're looking at a span of probably no more than twelve or thirteen years at most for him to go from a random nobody to a ubiquitous cultural touchstone, so well-known that tales of his supposed deeds are an entire genre unto themselves. Most, if not all, of the people who witnessed those events first hand would still be alive and able to recount the truth behind the legend, but for some reason they never do.
A bigger problem is that Kvothe himself is still around. Pretty much all people who attain truly legendary status only do so after they're dead, for good reason: it's much easier to mythologize someone when there's no risk of the mundane reality wandering in off the street to contradict you. But there's nothing stopping someone from tracking Kvothe down and debunking his entire legend...which, of course, is what Chronicler does.
Weirdly, the books don't seem to realize that this is what he's doing. The characters all act like Chronicler's interview with Kvothe will only enhance his reputation and status, even though whatever book Chronicler ends up putting out would mostly be an expose on how most of the things Kvothe is famous for didn't happen, and how he only became famous in the first place because the people around him were all superstitious, gullible fools. Somehow, I don't think that would go down too well with the Kvothe-addled populace.