Well, what would you have expected him to write? He was more clever than everyone around him in virtually every room he was in, for his whole life. Usain Bolt's biography would not have very many stories about the times he was outrun by people. Feynman had a healthy attitude about assigned-at-birth characteristics, he didn't deny them, but he didn't let them make him insufferable either.
Nonetheless, I see your point - most of the time, the alternative would have been to simply not tell the story, and I'm glad these stories were told.
If this is true, then maybe Feynman really was insecure about his intelligence ("humble"), and his "boastful" stories were to make him feel better about himself and to cover up this insecurity.
(I am not a psychologist, so I can be completely wrong here.)
However, whenever he talks about getting recruited or being tasked with something important, he always claimed to be astonished that anyone thought "little Feynman" worthy of such a high task. I take him at his word that this was his reaction.
I'm not sure what to make of this.
I'll add that though his bragging felt a little distasteful to me, I greatly enjoyed the tales and deeply appreciated his honesty and forthrightness about his life (including aspects I find less savory, like womanizing).
I'd absolutely recommend the book, but perhaps try _The Feynman Tapes_ on audiobook if possible because they're the recordings of his own speech that was written down and became the book.