If you can suffer youtube ads, I bet there's a bunch of video's on there showing people taking Fluke's apart for all reasons.
If you had another DMM, you might be able to trace the signal path from the probe, into some voltage divider network. Maybe some solder joint has broke around there. Or some resistor cracked ??
But there's probably lot's of SMD sized parts, and that can be tricky to probe on, without shorting stuff by accident. I usually solder on wires in a few places, and then use alligator clips, like for the ground probe.
Hopefully, the problem is outside of the chips. But yeah if I get to 77yo, IDK what my soldering skills would be like. Maybe by then I'll make a robotic soldering station.
Here's a service manual, for some version of the F77. Ok and you are an EE, I'm just a hobbyist. So that's good.