Finally got around to installing the replacement IEC inlet / EMI filter on my spectrum analyzer. Hit the power button and it powered up as it should, thank satan.
Then I ran a signal into it and the results were a bit... strange. It displayed the frequency way offset from what it should be, and there was some weirdness in what was displayed, and the problem was the same on both inputs. So I went into the self-test menu and most stuff was fine, but every test for the digital filter board errored out. Oh dear. It seemed fine and reseating it didn't accomplish anything, so I spent the next couple hours with the service manual, checking voltages, clock signals, data lines, etc.
Everything seemed to check out fine. Everything suggested it should be working. I decided on a whim to run the self-test again, and... it worked? Hooked it back up to a signal, and... it's fine now? Wat.
I dunno what the actual problem was, maybe one of the other boards I pulled in the process of checking things needed to be reseated, maybe it just needed to warm up for a while longer to shake off the cobwebs, who knows. All I know is it's finally working again, and all it took was a $16 part, so I'm happy.