Very insightful, thank-you both. I haven't made it back into the dungeon yet, but I will add those tests to my to-do list.
Regarding the 5V rail, yes, now that I take a closer look at the schematic--something I'm not terribly good at--I see that it is in fact TWO power supplies, and I AM capable of understanding how the 5V section works. "Not very well", indeed, it seems. I should also check those two diodes. One of them may be limping.
I am excited by the idea of replacing the 5V rail wholesale, and I think it's within my skill set. MED6753's modification is brilliant. I have bridges, a 2200uF, and some 7805's.
Back downstairs, where my wife occasionally throws scraps of food and a picture of the children,...
[Later,...] So. I tested the two diodes of the rectifier, and found them both leaky. After replacing them, the "5V output" was still 4.3V. I suppose that leaves C19 as the most likely suspect.
So I did an experiment. I "re-enforced" the 5V rail with an external 5V power source at C19. The "noise" level (with the Fluke's probes unconnected) dropped from a very sporadic something-or-other, to 0.001 at most, and when connected to my little utility power supply now registered an entirely reasonable 13.67 Volts!
So it IS a 5V problem! I suspect some logic circuitry is not being graced a healthy enough diet to function,... well, logically.
Now to fix. Questions come to mind.
Could use be made of one of those darling little $1 buck converters, like this one?
https://www.ebay.ca/itm/225433817428If I used the FULL winding of that part of the
existing transformer, there's about 12VAC there. I could bridge rectify, filter, then dial up about 7V on the buck converter, and add a 7805 for stability. Or am I getting too brazen?