Most of the individual constraints proposed in the myriad papers on GB and Minimalism are probably implementable by machine. But no one in Chomskyan generative syntax seems interested in explicitly spelling out the full set of principles that would underly a large-coverage grammar--except maybe Steedman's Minimalist Grammar formalism, which is ignored by most people who call themselves "syntacticians". In contrast, the HPSG and LFG communities have attempted to provide large-scale grammars and a lot of NLP work has used them in a serious way, but those communities are no longer very active.
Is there a work which lays out modern Minimalist generative syntax in full formal detail, and shows how this formal system handles a very large range of different syntactic phenomena? Such that it would be possible to produce a large hand-parsed corpus? It seems like this is what would be needed for generative syntax to have relevance outside linguistics departments. If it exists and I'm just unaware of it, I'd be glad to hear about it!
It's also a bit odd to suggest that any framework for which there isn't a wide coverage parser lacks any relevance outside linguistics departments. I know of lots of examples of cross-disciplinary work involving generative linguistics, but most of it doesn't relate to parsing at all. I'd say that wide coverage parsers are actually a pretty niche interest, which is one reason why people don't tend to work on them much.
The rest of academia has agreed on a set of methodologies and premises? No wonder we feel left out!
>increasingly ornate innate machinery is needed to explain how kids arrive at the 'right' grammar
This is too vague to respond to. But theories do tend to get more complex as we learn more, since there's more data to account for.
>where the structural representations were never great at predicting processing difficulty, nor implementable in a machine.
Which structural representations assumed by (say) GB theory are not implementable in a machine?
>he is IMHO pretty wrong in thinking that this latent structure can't be learned (e.g. inferring a probabilistic context-free grammar).
Chomsky always agreed that it was possible to learn context-free grammars via statistical methods. That's why he placed such a great emphasis in the 60s on showing that context-free grammars are not a suitable model for natural language.
Your last paragraph is fairly astonishing, insofar as it admits that generative linguistics has obtained lots of interesting results which cannot be characterized in "modern" terms. That sounds like a pretty strong indication that generative linguistics has got something right!