Good news, it's all fixed. But I'm not sure if it was broken originally.

So I replaced the switching transistor I blew up with ST BU508AW. I also replaced the 0.47ohm primary side current sense resistor, mains fuse and replaced all the electrolytic capacitors...
But it still wasn't starting up, so I put a scope on its output and found the rails did go to +5, +12 and -12, but only briefly before turning off. It also latches off, so unplugging isn't enough, the primary side capacitors have to be discharged before power is applied again.
I found optocoupler Z4 had over a volt across it, so I pulled zener diode D18, which was marked B3 12, and I measured it's reverse voltage as 12.8V. Which seems okay, except looking at the schematic I posted earlier (that I reversed engineered) it seems like Z4 might always have voltage over it? R27 is 100ohms which pulls one side of the zener to +4.3V, while the other side of the opto is at -12V, so it seems like the opto and zener would always be conducting.
I have no idea what that Z4 circuit is for or how it works, unless voltage over the opto is normal. I should go back and remeasure it now that it's working.
Anyway, none of that matters, because I decided to attach a HDD as a load, and the power supply worked! It must have been detecting a no load condition and latching off.
So I'm not sure why it didn't work originally, I think either one of the other components in the computer has a fault or the electrolytics were bad. I don't have an ESR meter, but they all tested roughly the right capacitance.

edit: I also have a second NEC APC III which doesn't turn on, it will be interesting to see if the power supply in that is faulty or not.