My nugdet of experience to add a data point to this claim. I have TMJ and have had it for as long as I can remember (my jaw clicks every time I open or close my mouth). I developed constant tinnitus in one ear (on the side where the clicks happen) in my early 30s, a few years ago. I was SURE the TMJ was causing my tinnitus, but the doctors dismissed it as non-related. Maybe I should have pushed harder. (I also have voluntary control over the tensor tympani muscle, for what it's worth.)

That the tinnitus is on the side of th TMJ is interesting.

Can you change the frequency/intensity of your tinnitus by changing pressure in the ear with fingers or by moving your jaw?

Not sure on the stats here but I’d speculate that a co-occurrence could be down to hEDS. It’s not as rare as doctors think and it can cause TMJ and tinnitus.