The question isn't "are neural networks just like human brains?" It is "can statistical models generate and understand language?"
The answer to the actual question under discussion appears to be a resounding yes. It does not appear that learning language is innate in the way that Chomsky has argued.
It may be that learning language as efficiently as human children will involve some innate property that we have not yet discovered. But that is a perfect example of moving the goal posts. We go from "there is something special about the brain that allows us to learn language, something that cannot be reproduced statistically" to "there is something special about the brain that allows us to learn language efficiently, more than statistical models".