Confirmed the speaker protection relay was bad. Found a reasonable replacement on Amazon.
Dong Woo DW321-D12S is the bad one. New one is a lot shorter and advertised as 10A but the text embossed on it says it's 12A, maybe it's only 10A for DC?
Here's the new relay installed. It's the thing that says NTE on it.
Also found the fan was missing on the 2nd amp. There's supposed to be an 80mm fan in front of the big aluminum heatsink all of those mosfets are screwed to.
After replacing the relay on the 2nd amp and powering it up, a capacitor blew on the right side input/graphical signal indicator circuit board, and let all of the factory smoke out. I replaced that cap, and had audio working on both channels, but the graphical display didn't work. I wound up just swapping that board from the first amp into the second amp.
I used the front panel and potentiometers from the first amp as they were in better shape, and I had already swapped front panel while trying to diagnose the graphical display issue and didn't see any point in swapping it back. The speaker level output board on the first amp was in better shape than the second one, so I swapped it too.
Without knowing any better I thought one of the wires on the board was being really stubborn to pull out, so I pulled harder. Turns out it wasn't in a socket on the board it was in a little plastic terminal permanently affixed to the board. My Optometrist has been trying to put me in bifocals for 4 years now, it may finally be time.
I improvised a new solution.
It all seems to be working now. The eBay seller has been great, really bent over backwards to help. Without the second spare parts amp I would have never fixed the first one. I guess really, I used parts from the first amp to fix the second amp. Regardless, dgd_surplus on eBay is good people!