Another theory I have is that is a specific IO pin that is shortcircuit and therefore draws to much current.
I've run into that on a DMM before, analog ASIC had been partly destroyed by someone abusing the autoranging Amp input. Mostly worked except for a few functions where battery draw shot right up...
Yes it is quite odd that everything otherwise works on your Fluke.
It could be a specific in or out pin that is shorted but this could be external or internal and when it's internal it's much harder to solve.
As this meter does AC+DC measurements the RMS converter is likely powered as soon as Volts measurements are selected, is the converter external? Does it get warm?
I assume you have checked the switch contacts? I doubt they'd get that dirty that they'd be a heavy leakage across them though.
If you have another Fluke with the same ASIC you could try comparative measurements to try to find the bad pin. My favourite method for this is the Huntron or Hameg or Fluke 867 curve tracer.
A more brute-force approach could be a carefully set lab power supply to up the loaded power rail, carefully watching with the thermal camera. (Just be aware that you could end up with it working less than it is... This approach can either just make things hotter, finish shorting them or even sometimes clear the short.)
I don't really believe in the leaky capacitor, that should have shown on the thermal camera. I could be wrong of course.