As much as I admire and respect Chomsky, would it be too far-fetched to claim that he lacks field work and expects the world to adapt to his theories, rather than adapting his theories to the world?
Theoretical physicists sometimes suffer from the same problem. They see reality as an approximation of theory. Experimental physicists are the ones "in the trenches", so why not listen to what they have to say? Same goes for linguists. Advances in theory should be based on experiment and observation. Otherwise, theory becomes l'art pour l'art and degenerates.
Field work for linguists amounts to working with people on dying and lost languages, but also the hard work of understanding and analyzing the intricacies of common languages like Hindi, German, and English. While Chomsky himself may not spend an inordinate amount of time proving his theories in the field, others have.
It should be noted, that Chomsky's theories have changed over time to reflect current findings. But the basic ideas of Universal Grammar and the essence of recursion in language have stayed the same.
As far as their culture is concerned, there is evidence from other anthropologists that the Piraha do indeed have creation myths (the rebuttal paper actually includes one on page 44).
People jump on stories like these because it makes for a good news story ("Rebel Linguist Takes Down Noam Chomsky"). But the evidence I've seen doesn't sound so convincing.