It may be that calculus does not provide a description of all kinds of motion, but fortunately it does for most of the ones we encounter in everyday experience. Also, it might be an example of "looking for your keys where the lighting is good." Motions that are differentiable are easy (relatively speaking) to analyze, so we analyze them.
For instance, it seems to be the case that large-scale motions are approximately continuous with continuous first derivatives. That is, there are no infinite accelerations in the real world of large-scale things. For infinite accelerations you need an ideal world where there are, for example, perfectly rigid billiard balls that can experience an instantaneous transfer of momentum from one to the other. But it is possible in the real world to have discontinuity of acceleration. If you roll a flexible hoop down a circular ramp that straightens out at the bottom, the hoop will uncompress and hop off the ramp at the beginning of the flat part of the ramp, even though the ramp itself (and its slope) is continuous the whole way. So large-scale motions are well approximated by differentiable (and thus continuous) functions, but generally not by infinitely differentiable functions.
But as to whether continuity is an necessary property of motion, I don't know. It might be that it is only a good approximation over a very large and useful range of scales, just as the continuous medium approximation works well for fluid dynamics until you get down to the scale of individual molecules. If motion weren't at least approximately continuous, I don't know how we would ever establish that the thing that was here is identical to the thing that is now there. Experience would just be a blur of unconnected events.