Basic vector addition lets us visualize that a parallelgram grid is formed from the displacement contributed by initial velocity and displacement contributed by acceleration. So it's pretty easy to see that the middle control point forms the direction of acceleration with the midpoint between start and end[1].

Why then are all the sources describing quadratic Bezier curves make absolutely no reference to any meaningful physical quantities of any sort, instead just treating them like some sort of magical incantation about "interpolations of interpolation"? And similarly cubic Bezier curve can easily be described by the parallelpipid grid spanned by the velocity, acceleration, and jerk vectors, so there's no sense in saying that this strange way of describing them is necessary for generalization. The only likely explanation I can think of is laziness, as it doesn't really make much sense to be motivated by some sort of elitism on something so exceedingly simple.

[1] https://i.imgur.com/vP9fLYh.png