Wow..hmmm. That sucks. You may have hit the nail on the head with the "predecessor" thing. Sounds like your in IT or something. Or at least acting in that capacity for friends/family. But that DOES NOT sound like fun.
My only advice I could give you, is NOT very good. Hopefully someone MUCH more knowledgeable than myself will chime in here and offer some help. Shoot, you yourself probably have more tech know-how in your little finger.
If i WAS in your situation however, and ALL i had was my very limited hardware/software knowledge. I'd probably be doing an obscene amount of
intelligent trial and error. Keyword in intelligent. lol. Like i would literally grab one of the "good PC s" and then one of the "bad" ones. And then (very inefficiently) take a SINGLE part from the "bad" and install in the "good", and then see what happens. Literally one at a time. Ram module to ram module..psu to psu...Rinse and repeat. But i like said. There HAS to be a more efficient way of troubleshooting. I wouldnt know unfortunately. And it kinda sounds like you may have done that already. Sry sir. Good luck to ya
Edit:
Also...may also want include more details as to what not "staying up" means. I get you. Just not very specific. For instance, does the PC post...boot into windows (or whatever OS ur running)...does it sit idle at desktop screen w/o issue. Or do the issues only manifest under "load"? Or even a specific program? Im sure this info would help any other TR members who wanted put their two cents in.
Love the sig btw