The subtle thing

OK, the center of mass moves like a point mass (= total mass of a compound object) under the influence of external forces only.

Now consider a non-center-of-mass point of the compound object. Does the external force have any influence at all on that point? The answer is Yes, Yes, Yes! In general, any point of an object is under the influence of external and internal forces. The center of mass happens to be where all internal forces magically cancel out! It does not mean that external forces do NOT apply to other points.

However, since it often suffices, or it is the most important, to figure out the motion of the center of mass, we are used to reducing a problem involving a complicated compound object to a point mass problem (without forgetting an internal rotation or an internal motion, if applicable), but we should never forget that external forces act on other parts, and the center of mass feels external forces through those other parts around it.

Just a subtle point to make clear, in order to avoid any misunderstanding. Maybe it is already clear to many students?!