This is simply untrue. A projector is the very definition of an intermittent motion system. As you yourself say, the way a projector works is by momentarily holding the film in the gate. Without that still image, persistence of vision wouldn’t work, and we’d see streaks, not a moving image. And if it was as unsteady as you say, then in projection you’d see the frame moving (not from frame to frame, but while a single frame is up), because the projection will dramatically magnify any defects. This just isn’t a problem.
The time it takes to make an exposure is measured in single or double digit milliseconds (depending on your optics and light source of course). In the case of our camera, our typical exposure time is 8ms, which is 1/125th of a second. That’s significantly shorter than the time the film is held still in the gate during projection.
Rolling shutters become a problem if you’re trying to capture something that’s actively in motion. they will work fine in a projector-based system if the exposures are taken during the part of cycle where the film is held still. It is definitely held still enough that a rolling shutter will work, and even if there is some creep of the frame through the gate during the still phase of the cycle, if your exposure times are short it’s not a problem. And that’s not hard to achieve with inexpensive high-powered LEDs.
Where projectors vary is in the stability of the image from frame to frame – some will have better edge guiding and pressure plates that will hold the film steadier. Others will not, and from one frame to the next you might see some variation. But that’s going to be the case whether you have a rolling or global shutter and it’s a matter of post-processing that you’d likely be doing anyway, to make things stable later.