I've never run across this problem before and I don't recall reading about anyone else ever having it. Online searches haven't helped much either- apparently it's a very rare problem. I tested the original computer with a known good PSU and when it wouldn't start with that, I mistakenly assumed that the MOBO was the whole problem. Since we are short on spare PSUs and cash around here, I wanted to save the original PSU for other systems and wanted to make sure it worked before I saved it.
Using 20/20 hindsight, I did a multitester pinout test and one of the +5 volt pins read about 8 volts. That is several volts above tolerance and is probably what ruined the boards. I'm going to have to be more careful in the future. Fortunately, the systems I ruined are older systems that already had problems so it's not a huge loss but I would've liked to be able to fix them. Now that doesn't look like a feasible option.