Linguistics is the one thing he actually knows something about. His theories have been a large influence on computer science as well.
The essence of the problem with his work is the nominalism of "deep structure". His hypothesis is essentially the homunculus hypothesis, the little man inside the man who understands deep structure. He has sort of dropped that hypothesis but still refuses to acknowledge that grammar is part of learning and understanding language, not some innate capacity for grammar.
I studied the Chomsky theory taught by a Chomskyite (also a leftist) in graduate studies and tried to make it work. It didn't.