"I am guessing a problem in the power supply (there's a little experience guiding that), so I dug in and opened it up."
I thought it's dangerous to open those power supply boxes, or is it not?
Step number one. Power down the unit. Stare at the disconnected plug in your hand for several seconds.
... Step one A -- see below.
Step number two. Know what capacitors look like and avoid them. They bite!
Step number three. Touch nothing inside with your own flesh.
Step number four. If you don't have the tools, materials, training or patience to follow these rules, go to step one A. -- Throw it out and buy another.
You have to know what you're dealing with. In this case, to begin with, it's a 1979 Amana Radarange. In 1979 switching power supplies were relatively rare -- and in any case had it used a switcher my job would have been done, because the symptoms were "consistent with" too much AC ripple on the DC power supply to the CPU chip, and it would take a lot more knowledge to troubleshoot the supply. And it's the switchers that (can) develop the really high voltages, so I probably would not have even bothered.
What I expected to find (and did!) was a simple step-down transformer whose AC output was rectified, then regulated. My guess was that the "averaging" capacitor was failing; it's been 26 years, after all, and electrolytics do not last forever. So I was dealing with voltages in the range of 7 to 10 volts, no more. And it was all on an open circuit board, though getting that out of the extruded frame was a bear.
You can see it in the photo below; the power supply is the board pulled out of the frame. The two capacitors to be replaced (< $2.00 total) are the two blue cylinders.