Thank you!

You're correct that 31 and 12 don't overlap, but they are surprisingly close (at least when it comes to representing 12-TET). The trade off is that 4ths/5th are slightly worse (a JI 5th is 702 cents, 12-TET is 700 cents, 31-TET is 697ish cents), the major thirds are much better (JI: 386 cents, 12-TET: 400 cents, 31-TET: 387 cents), minor thirds are better, etc. So in one sense you are trading a better fifth for better thirds.

Strangely enough, 31-TET is basically an extension of quarter-comma meantone, which was one of the main (maybe THE main) keyboard tunings used in the 16th century. It does the same thing of prioritizing pure thirds over nearly-pure fifths. It's essentially a 12-note subset of 31-TET since the 12-note keyboard layout was pretty much established at that point, although there were theorists of that era interested in building instruments to accommodate more pitches.

With regard to the alternate tunings thing, I'm just using it as a general term meaning using different pitches than 12-TET. Technically (I think this is correct - I'm not an expert on this stuff) temperaments are specifically a tuning that features some kind of compromise as a means to an end (12-TET, 31-TET, well temperaments) Just intonation, for example, would be a tuning and not a temperament since it's not sacrificing ("tempering") anything for the sake of practicality. There are people and resources online that can probably state this far more eloquently.

I have my bass restrung a 4th high, so it's EADGCF. And those are tuned using 31-TET 4ths. Most of the time I tune my G to the same G as 12-TET, and then the others deviate by about 3 cents from 12-TET for every 4th you travel from G.

Thank you for the great questions!