At distance, I can only use my experience and some statistics. But I can't do a thorough diagnosing. Bad grounding may make worse any existing problem.
If you see often the need to retune the instrument, grounded or not, there is definitively something wrong and you should try to find out the cause instead of repeatedly curing only the symptoms.
The partially broken 5mH coil is a common source for such problems, another is sometimes a partially defective diode D1 (which may look fine with the multimeter but the oscilloscope will show a breakdown at ~20V in reverse direction). In rare cases I have also seen ghost tones coming from the circuitry around the volume tuning stage (Q8 and friends) which started sometimes oscillating itself and which gave an unwanted amplitude modulation of the audio signal.
I know that Thereminists are often in a difficult situation. There are only a few competent service points around the world and money is also often an issue. But in some way you should see your instrument like a car which needs to be serviced in fixed time intervals or after a number of kilometers or miles...
BTW: I plan to improve the general situation starting next year and to offer weekend classes for people who want to learn how to service and fix theremins (semi-) professionally.