I'm not sure why you're asking this about Minimalism specifically. There are already wide coverage parsers based on various generative frameworks (e.g. HPSG, LFG, CCG).
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.
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.