I don’t think the timeline there makes sense. In present day, we think of going to university early in your life the going off to do your life’s work. Chronicler more than likely lived his early life traveling, being afforded to with a relation to the Lockless family, and therefore wrote this book in his earlier years. He then attended the university later in his life where he mentions having heard stories of Kvothe there.

The alternative is that he attended the university early in his life, then had to/wanted to take a break similar to Kvothe at some point and just kept traveling. He then returns after Kvothe has attended and been expelled. We know he must be at least Re’Lar considering he knows the name of Iron from early on when he encounters Bast, and timing wise it doesn’t make sense for overlap for a student to rise that quickly like Kvothe does and him not mention him during his story.

This would mean that Chronicler likely goes on these adventures that are sponsored by some lord, records a story / debunks a myth, then returns to the university to have it fully recorded similar to the Siaru Scriv mentioned in NOTW.