Anonymous, you put that in the clearest and most excellent manner. My partner and I were working on this today, and your post said exactly what we both needed to hear. I've even forwarded it to him.
Of course, you can overdo the CBMP if you're too gung-ho about it. We found that out today...we were practicing setting up and then swinging our first half natural in the waltz, and he roated so much in the set up that I could feel the tops of his thighs starting to cross. Naturally, the half natural ended up looking and feeling very un-natural. We tried it with less rotation through the top the next time (but with more than we usually used), and discovered what you so cogently explained above.
So thanks for putting it so well!