Hi Zeitkind, welcome to the forum,
Yes, I can hopefully help you. I have a 1041M (mil version) a 1045 (DC only) and a 1030A (RMS) all of which use the same A/D board.
https://www.eevblog.com/forum/chat/show-your-multimeter!/msg793266/#msg793266Your 1041 is definitely worth persevering with, it has very decent accuracy for a 4 1/2 digit meter (0.01% +1dig DC), a 10mV range with 1uV resolution and the same true RMS converter as the later 106x and 107x ranges (assuming yours has the AC option fitted). The best thing is that it is completely discrete (no micro, s/w or Cal RAM) so you can maintain it almost indefinitely. Nice Panaplex displays too. You can easily change it to display 10 readings/sec too. So don't bin it!
Discovering and replacing the Mains filter is a good first step
, as is checking the three supply rails - the tantalum beads on the outputs of the 15V regulators are also prone to failure. The axial electrolytics seem to last indefinitely so no point in changing them, as you have found. Good that it is displaying something too (+180V, drivers and displays functional).
The 104x and 105x do share most boards (apart from the A/D and display board, there are differences in the quality of ranging resistors etc on the other boards (higher quality on the 105x) but they are interchangeable. Does having the 1051 schematics also mean that you have a 1051?
All the logic ICs are socketed, which will help your debug. The clock oscillator is a simple transistor astable (no pll complication like the 1051), but I suspect that the problem is in the general triggering logic. What colour IC sockets does it have? The translucent poyethene looking ones are less reliable than the black ones.
As the schematics are only available in paper form I will need to get the camera out. In the meantime it would helpful you can post some photos to judge the general condition see if there is anything obvious. Also post which options are fitted.
One other thing, check that all buttons (hold, print inhibit etc) in the centre contol group are not pressed in!
P.S. Does the front panel say 'datron', 'datron Ltd' or 'datron instruments Limited'? This helps to judge the age.