I took a gamble on a cheap ($220) Metcal MX-5200 soldering station on eBay that was listed as only tested for power-on. The unit did power on but gave an open error for the left channel. Something was rattling inside so I opened it up and found two of the LCD screws and nuts were rolling around, seemingly from someone trying to open the unit previously. Uh-oh.
Factory resetting didn't help. I dumped the ROM since I saw another post here saying the ROMs can become corrupted. It seemed fine. In case it's helpful to others, see the attachment for a dump of 1.3.6.
Next I figured I'd try swap testing things between the two channels. I started with the relays. After swapping the left and right relays with each other, both channels started working fine! It now switches between channels and runs them both at the same time without issue. I didn't notice any solder issues prior to the swap, but perhaps there was a cold joint. Is there any other reason why swapping them would help?
The only problem left is a broken power switch. It seems to be a Cherry RR DPST 16A 125V part number RR812C1000-116 which is on DigiKey.