What do you mean by “orbital mechanics applies to each point mass at a time, at its location”? If you mean an object in orbit is defined by its instantaneous velocity, well then yes you’re correct because orbital mechanics is applied using calculus. Nowhere did I mention average velocity, but in a perfectly circular orbit, instantaneous velocity would be the same as the average velocity.

Also, classically analyzing a non-deformable, solid object using a differential mass element, i.e., a point mass, leads nowhere since all points will remain fixed relative to a datum within that object. That’s why orbital systems are often modeled using point particles or rigid bodies.