Work has resumed.
Initial testing was to see what the voltage rails were like without any load. +5, -5 and -12 were in spec but +12 is only 11.2v until I attached a hard drive and that oulled it up to 11.8v (still low but just within limits). At the very least though the PSU is behaving well enough to resume testing.
Starting with the backplane I found and removed two shorted tantalum capacitors on the -12v rail, clearing the short we had previously found. When reinstalled however all but the +5 was coming up immediately, until another tant that had not been reporting itself as shorting blew itself apart. I checked all those tants. Why did you decide to still give out?? Removed that one and tried again. All four voltages came up but suddenly -12 vanished again. Sure enough, another tant on the board had shorted out. I'm amazed the PSU is not kicking out with these near 0 ohm shorts. Removed it AGAIN, let the PSU run with the backplane and a load on the power rails and so far things are stable so I think we got them all.
I only briefly checked the three ISA cards but it appears the SBC and VGO boards also have ground-to-rail shorts. I'll deal with those once the backplane stops screwing with me.