Hi Thilo,
That is a really nice and sturdily built scope, and it is also a bonus that you can power it from a battery!
That red square device is a ceramic disk capacitor, 33nF, low voltage, used for bypass purposes (not tight tolerance).
BTW, that 50k trimpot, at this age, is probably not a high reliability component any longer, I would replace that too!
Those old electrolytic capacitors you mentioned, check them for ESR as well, not just for capacitance!
If you get some voltages out from the inverter, then it must be going, probably something is loading it down.
Disconnect those E-series rectifiers, those could very well be the components loading down the inverter.
Instead of looking at that 20mV calibrating signal, look at the signal at the collector of T201, T202. That would tell you how is the inverter running.
You could probably test the unit out of the scope, just provide 10V to the "G" and "M" (24, 23) points.
You need to provide another reference point, "B" (29). The schematics mentions that it is at -10V potential, not ground, though it does not show how does that get there?
The high voltage rectifier diodes (E700 type) you can replace with any 1.5kV diode (or possibly 2kV, in case the inverter is running without any load, then the voltages could be higher).
Just plug in the parameter into the selection guide of your favorite electronic supplier over there. I do not think that speed would be an issue at 15kHz.
For the other diodes (E400, E600 type) simple 1N4007 should do.
You also might want to check for leakage of those 10nF filter capacitors in the HV voltage doubler. Again, there you would need capacitors with about 1.5kV or higher rating.
On second take, having looked at the picture of the circuit board, those are styroflex capacitors, those are most probably fine, the paper tube components are the "HV diodes".
Those E-series HV diodes seem to be stacked rectifier components in those paper tubes, not a single rectifier element, like a diode, with a single junction.
To check it out you would need higher voltage, a DMM diode checker does not provide much more testing voltage than that is enough for a single junction.
Those are also probably not silicon diode junctions either, but possibly selenium. Again, those need higher voltages to check.
Looks like the part number (E400, E600, E700) could mean the number of elements in the unit (4, 6, 7, the length of the units are different).
Use at least 100V to check them. Do check the leakage current, that is what could be loading down the inverter.
Regards, Peter