You can use Clean 'n' Dull on a spinner, but you have to make sure you don't have the speed set too high and/or use too much pressure. I HAVE seen the results of a ball getting a glaze of C&D burnt into it, and it's not pretty. The ball just completely labored in oil like it had an extender polish applied or something (but this was a clean towel with nothing but C&D on it). A little fresh surface took the glaze right off and the ball came back to life, but still.
And yes, the new C&D bottle's instructions no longer mention using a damp towel to wipe it off, which surprises me. I still use the damp towel step and it works very well that way.