Just to clarify a few things that may have gotten lost in translation... First, I am definitely doing as Sensei instructs and practicing it that way, because I understand the importance of obedience to an instructor and of respecting his years of experience and expertise.
Second, when I speak of pivoting on my heels, I certainly don't mean to suggest that I lift the rest of my foot upward and loose all contact with the ground except for my heels, wheeling around like I'm on the stubs of my legs with no feet at all. I only mean that the heels are the center of the pivot.
The reasons that seems helpful to me are (a) the construction of my legs places the balls of my feet of typical allignment with the rest of the leg and body, so using them as my balancing point requires some odd contortions of my hip and knee joints; and (b) I have next-to-no mobility in my lower ankle joints, and perform most pivoting action actually with my tiba and fibula. (Don't try too hard to figure that one out.
At any rate, I am continuing to practice the ball pivot, hoping that training might lead me to finding some new muscles and tendons and call into action that might end up helping out. Thanks again for all your various perspectives on this, and I must admit a little regret at the hostility that the question seems to have brought out of some.