Excellent! Thanks for the pointers, Castellen.
So, I just cracked it all back open, powered it on, and probed at it with my multimeter. The bridge rectifier seemed to behave as I'd expect four diodes to behave. Everything worked in the right directions. I powered it on, checked voltages on it, and everything looked reasonable. I've never worked on such a thing, so the +80 and -80 volts coming out of it (with 110v AC going in to it) seemed somewhat reasonable. I had expected it to be +-60 volts, but whatever.
There's a line printed on the board by the smallish transformer there, so I assumed that on one side it's AC, and the other side is DC... I checked voltage from ground to the heat sinks on the two transistors on the AC side, and was getting 80v, which was interesting...
I did notice one of the ICs on the DC side had a bit of dust on it, that I neglected to remove last night, so I did that. powered it back on, and continued probing. I had the black lead of my multimeter jammed into the black (ground) pin on the motherboard connector (not connected to the motherboard), and I started probing around the DC side. I was getting 0v on most of the chips, then i tried one, and got 5v. then i checked the cable, and I was getting all expected voltages again. +5, +12, -12, ~3v (the TICK pin). I plugged it into the motherboard, turned it on, and it powered up.
I reassembled the whole thing, turned on the switch and.... nothing. No red LED anymore, nothing. I was getting frustrated but then I just decided (for whatever reason) to leave it on for a few minutes, to see what happened. After a minute or so, it came to life. That was about an hour ago, and it's still on without rebooting or powering off. I recopied the Kickstart floppy that failed a moment earlier (I think I put my magnetic screwdriver on it. oops). As an A1000 owner, you can never have too many backups of Kickstart floppies!
I'll check those resistors in the future (I meant to do it, but then it came to life and I forgot) for the next time this happens. I'm getting really quick with opening this thing.
So... do I trust it? Probably not. I'm still looking into my options, but being that I didn't really have the cash to do anything about it, it's nice that I got a bit more (admittedly, borrowed) time on it.