Thanks kindly for all of the replies to my previous question. I've read
twice through each reply - some will take another run-through tomorrow to
really absorb, since I do want to understand it all completely

But, I just did an experiment which tells me that the PS is probably not
the problem. I borrowed a Compaq and pulled its working ATX PS. The
resulting behavior was the same - power on for a moment only.

Does this pretty much say that the PS is not the problem here? What
should be my next step in diagnosis?

From the reply by w_tom1, it seems I must check all the pins. Tom says to
check for shorts - but I have only a 9v analog powered multimeter, so I'd
think I can't use the ohm meter, except on 12v circuits, yes? Or do I use
the voltmeter and look for what pins stay mostly flat on voltage and
which pins might spike? Or does the analog meter (an old Radio Shack one)
respond too slowly to tell me anything?

Thanks again for all replies.