Who would have thought it would be so easy?
A little attention to the reset switch and suddenly we have video and sound, loud and clear.
That means the reset mechanism was the actual cause of the no-video, no-sound symptom on this console.
I still decided to replace the electrolytic capacitors while I had the machine apart.
That is maintenance, though, and should be kept separate from the diagnosis.
The fact that a capacitor is twenty years old does not, by itself, prove that it caused this particular fault.
Still, one apparently dead SNES is back among the living.