Pretty sure he didn't. Also, Everett book isn't only about the controversy with Chomsky, it's about his own experience in the jungle and his coming of age. It's a great book even if you absolutely love Chomsky.
He did too. Everett tries to make the point that because Piraha doesn't use recursion (which we only know by his own account, and nobody else's, since nobody else knows Piraha as well as he and the Piraha do, also by his account) then Chomsky must be wrong.
How is Chomsky wrong if Piraha doesn't have recursion? According to Everett, always, Chomsky's position that recursion is the distinguishing characteristic of human language, must necessarily mean that all human languages have recursion.
This is exactly like saying that, because it only snows in winter, it's not winter if it's not snowing.
For Chomsky to be right, it suffices for a single human language to display recursion. If even one human language displays recursion, then humans in general can learn any language that displays recursion- because we know well that human infants learn the language of the linguistic communities they're reared in (and therefore any human can learn any human language). For instance, even a Piraha baby raised in a Brazilian family would learn to speak Portuguese, and Portuguese has recursive embedding.
Everett of course claims that Piraha, somehow magically unlike any other human being in the world, are incapable of learning any other language than Piraha. He also claims that they were unable to learn simple arithmetic, beyond 1 + 1 = 2, despite his, um, best efforts.
In fact, all human languages except Piraha, and only by Everett's account, display recursion. Which makes Everett's claim about Piraha so hard to accept. The fact that he remains the only (self-professed) authority on Piraha makes it even harder to take him seriously.
Generally, it's not so much that Chomsky has won anything here. Everett is so clearly a total troll, and his books the printed equivalent of clickbait, that it's ridiculous to even claim there is anything like a debate to be had. It's like "debating" a climate denialist.
"Don't sleep, there are snakes" which recounts the experience of Everett among the Pirahas people is a fantastic book.