Hi Christopher
It was awhile ago now but from what I remember, I boiled a pot of water and then turned it down to a fast simmer, turned the oven on (the oven has to be on for the meat probe circuit to get power) and then with the probe connected I would dip it into the hot water and watch the temperature gauge on the control panel, take it out for a min and then put it back in, repeat. Eventually I saw that the temperature gauge would start moving, very slowly at first and after several cycles of this, it started moving at the pace it should have been moving when placed into the hot water. When I got the stove it hadn't been used in at least 50 years so I think it was just a bit stuck.
Before I had done this, I checked continuity between the plug and the gauge so I knew it wasn't a loose or broken wire/connection. I believe there is a transformer in the circuit to step down the voltage as well so you would want to check that too.
You are right, there are some detailed tests in the service manual but I didn't end up needing to go that far. Ever since I did that its worked perfectly. I use the probe anytime I'm roasting meat and find it super handy and has been spot on accurate to measure doneness correctly. I know I'd really miss having it so hope you can get yours working.
Hope this helps, let us know how it goes. Feel free to message/email if I didn't explain anything well.