I don't know any more about camera motors than what I just read [1], but yes, the quartz crystals keep the motors running at the exact same speed. This kept the film camera and the magnetic tape recorder synchronized.

A semi-related thing that I find interesting is the clapperboard that film makers use at the start of each scene they record. I had assumed this was a cliche tradition, but it's an effective way to synchronize the film and audio. You can see on the film the frame where the clapper closes and easily synchronize this with the sudden loud noise in the audio.

[1] https://www.filmmaking.net/filmmakers-faq/130/what-is-crysta...

This works for short recordings, but for longer recordings explicit continuous sync is needed, almost universally using Timecode, because normal quartz oscillators are not precise nor stable enough to guarantee single frame accurate sync over, say, an hour (which would be 10 ppm or so, 1/3600*30)