No it would not. The money necessary to find a lab is orders of magnitude higher than any reasonable UBI.
In my case it's both, so instead of teaching I take half-time IT consulting gigs here and there, which pay enough for me to be able to do science even if I had no funding at all. I still get some grants, mostly because you're looked at funny if you don't, but I'm done losing sleep over them, plus if I finally decide to give up academia I'll have an easy transition.
While teaching helps consolidate your own understanding, and is useful and satisfying, it's a matter of degree.
What is really expensive and complicated for now is “wet” data collections: it is a finicky process stuffed to the brim with very expensive machinery, consumables, lab space and manual workers. Computing power is not really a limitation nowadays, you can comparably get a whole lot of bangs for your bucks.
Concerning the sharing part, equipment pooling is definitely already happening, at least at the local scale. And what is getting more and more developed is the “platform” concept, where some labs/teams slowly transition from doing research to producing data according to a standardized protocol (genome sequencing, genotyping, ChIP-seq, etc.) for other teams doing the research.
I noticed companies moving into the space, like 23andme, and wondered if they might work in combination to keep consumer prices high...
23andme is not doing sequencing, they're doing genotyping. And they are doing so using relatively old and well-established method; so I highly doubt they will be a force of innovation in the sector for the coming times.
Also, engineering leads science at time: this works but we don't know why.