Yay, the power supply is fixed!
Thomas/LA3PNA suggested I look at the input and reference voltages going into the μA723 regulator, so while I was at it I looked at all the pins in the empty socket. The only unusual thing I found was about 0.1V where the output pin of the regulator went. I’m guessing the voltage regulator probably didn’t like having that much voltage on its output pin, and that’s probably what was killing them.
Pin 10 went to the anonymous red transistor type thing, which in turn was connected to the pass transistors and Vcc.
Consulting with Thomas again, he said it was probably a Darlington transistor or something similar that failed. The μA723 output turns on this transistor, which is then able to provide more current to turn on the pass transistors than the μA723 alone would be able to.
Took the transistor thing out and connected the μA723 output to the base of the pass transistors and everything worked! Got a stable 13.7V at the output of the power supply, and I could turn it off and on again without any problems.
Found a TIP31 power transistor in my collection of parts and put that in place of the dead red transistor and it looks like this power supply is back in business.
It’s a pretty ugly soldering job, but I think it will hold up. Still need to test it under a load though.