I finally got enough spare hours to do the repair.
First thing, I did replace the socket, and can testify that the "exact replacement for Blackface Deluxe" socket, isn't. There's a shield turret in the middle of the lugs on the bottom which makes it impossible to bend the two cathode lugs together the way Fender did (so I soldered in a jumper).
More important, the holes in the mounting ears are not the same size as the originals, which were attached with self-tapping (#6?) screws. Instead, they're tiny, requiring #2 screws, which I swiped from my collection of cartridge-mounting hardware. Exact replacement, my eye.
Anyway, once that was done I connected the socket up, using all new wires from the tag-board. While I was in there, I replaced the 1M resistors and the 22k resistor, both of which were looking pretty ratty.
Then I put the original tube back in, and got very different voltages on the plate, indicating severe unbalance. They were also on the high side, indicating that the tube's output was weak. I replaced it with the only other 12AT7 in the house, which at least had balanced voltages (well, roughly) on the plates. It still was weak, but at least I got results.
So I buttoned the amp up and played for an hour, and it didn't drop its gain once, that problem seems to be fixed. It didn't sound all that great, though; no real balls, a classic symptom of a weak driver tube in a Fender. I'll get a good 12AT7 next time I order something.
Thanks, all, for the ideas and support! I still don't know what the original problem was (my guess: wire with an internal break), but it's fixed.
Peace,
Paul