Thanks to Dave for giving me the troubleshooting tips I'm using to try to do this.
I have a Tenma 72-410A bench multimeter I got for cheap. Everything works on it but the display (by which I mean I can select the appropriate modes and have the LEDs light up, and it will also beep if I set it to continuity). So I just want to run what I did by you all to see if I'm on the right track.
The first thing I did was check the power rails. There are three regulators on the board - a 7805, a 7815, and a 7915. The 7815 and 7915 appear to be running from opposite sides of a bridge rectifier. I checked the 7805 and 7815 output, and everything was nominal. I checked the 7915 output and it was hovering at 8.5 volts.
Ah! I thought. The -15V rail isn't behaving right. So I took another power supply and plugged 15 volts into the output, being very careful to make sure it was the right polarity.
Something was drawing 1.4A, and I think the either I fried the regulator or it was already fried and I just made it obvious, because now it's producing 0.5 V.
In order to test further, I found the diode that protects the -15 regulator from stray voltages, and I pulled off one side of it, thereby isolating the -15V rail from the rest of the board. I also replaced the input filter cap and the output filter cap (the compensator cap is a surface mount and I wasn't sure what it was). The regulator is still putting out 0.5V. And it gets pretty hot.
So I just ordered a batch of 7915 regulators and am going to replace that, and see if it will give me 15V. Then I'll hook the diode back up and see what happens. If it doesn't give me -15V with the diode pulled out then it's something in the power supply.
Thankfully it's a really easy power supply...
Does this sound like a sane troubleshooting strategy to you all? I'm really pretty good at taking things apart and getting better at building them from scratch, but troubleshooting is something I'm still struggling with.