The “23.976 fps” frame rate is a direct result of video frame rates established when color TV was developed. Doing the 3:2 pulldown process (used to convert film to video), and matching the NTSC frame rate of 29.97 fps, we get:

24 * 29.97 / 30 = 23.976 fps

This rate did not result, accidentally, from the imprecision of old mechanical film cameras/projectors. You are right that 24 fps was chosen for film, but, as you say, the old cameras did not have the kind of precision to all be “actually shooting at 23.976.” It was a result of the chosen NTSC TV scan rate. So my point is that old film cameras shot at 24.0 fps (or rather as close to that as mechanically possible: +/- some error), not precisely 23.976 (which is used for video and derived from the formula above).