When mine did this, it was because an index wheel pivot was stuck in its jewel and twisted off. They are just like balance staff pivots, running in hole and cap jewels. I did not find a way to separate the cap jewels, but was able to clean them with ultrasonic.
Mine would run briefly, but the mechanical drag was enough to stop the fork. It would actually run then stop, then with no drag on the fork it would start humming again. Of course when it restarted, it would keep humming and not running because there was nothing to spin up the index wheel. I mistook the momentary stoppage for an electronics problem at first, but it was purely mechanical.
When I fixed the mechanical problems and serviced it, it worked fine. Mine had enough pivot left to clean up and work when I adjusted the endshake, but I expect to have to make a new staff (and maybe new jewel settings) for it one of these days.
If you disturb or unmount the fork, the timekeeping will be way off. If I remember right, the correct rate is 300 Hz. I got it close with my oscope's timebase, and then my Seiko Quartztester that I use for Accutrons would read it.