It demonstrates that human languages all have a certain structure, so it must be an innate faculty of human beings (as opposed to acquired). A neural network could not tell you that. It also suggests further avenues of inquiry (Why that particular structure?)

I think you misunderstood my question. Here's my understanding of Chomskyian linguistics:

1. You have a certain concept you wish to express.

2. You apply a generative grammar to the concept, producing a linguistic statement.

3. You express the statement in a linguistic performance.

4. I perceive the linguistic performance.

5. ¿I reverse the generative grammar to produce the concept?

6. I understand the concept.

My understanding is that Chomsky is only interested in steps 2 and 5 (and that he is explicitly uninterested in 3 and 4). But how does step 5 work?

I don't think much is understood about how linear externalizations of language are deserialized into symbolic structures, what those structures are, and how those are symbol structures are mapped into mental representations.

Hah. No pun :)