Another update: I decided to have a closer inspection at the solder
joints on the backside of my ZipGS, so I used my flatbed scanner as
a sort of digital magnify glass.
I spotted what appears to be a bad solder joint on one of the lower
oscillator socket pins (by no coincidence, this is also the area where
when touched, will cause the cache LED to flicker on and off). It looks
like all the solder around the pin dried out and is missing.
Here's a shot, showing the backside of the oscillator socket:
http://www.digitalsparkles.com/student_a/ZipGS1.jpg
Here's an close up of that suspect pin:
http://www.digitalsparkles.com/student_a/ZipGS2.jpg
My only concern is why pressing on it (to temporarily rebridge the
connection) still leaves the GS in a dead state. I'll try it in another
motherboard tomorrow but I think getting someone to resolder this
pin will be the first step.