I'm curious if there's new physics underway that may make neutrino interactions at a large scale feasible. Experiments that I know usually detect one to five neutrinos a year and I'm wondering if we could ever come up with method that can capture orders of magnitude more than that

It's not quite so bad, with many experiments looking at solar or atmospheric neutrinos seeing hundreds to thousands of neutrino events per year. Experiments at particle accelerator-based neutrino beams can see much larger event rates despite the small cross section, and as one example, the upcoming SBND experiment (https://sbn-nd.fnal.gov/) will see 7 million neutrino interactions in about three years.

Would need to make ultra dense materials which would require subatomic engineering.

Would such a thing even be attainable on earth without the tremendous forces that collapse atoms into neutrons? Does matter in a neutron density state convert back to atomic density state in the absence of such forces?

I think it would be unstable in general but perhaps some shielding could be created to help but even so it seems like strong forces would need to be used to hold the ultradense target together.