I don't really think about turning on balls of feet or heel anymore, but if I were to guess I'd say the turning is happening more at the center of the foot as a natural extension of the whole leg, or, more likely the whole foot turns (the load is spread on the whole foot).
How is this possible? Broadly speaking, your foot has 2 points of contact with the ground, your ball and your heel. It is easy to pivot around a point that is in contact with the ground. The middle of your foot, however, is not in contact with the ground, and turning around this point necessitates sliding the front and back of your foot simultaneously in opposite directions. I just tried it, and it's not completely impossible, but it's very difficult and puts a lot of strain on your knees while being slow and inefficient.
I understand the stepping thing when you are talking about actually moving your feet, but what about when you are turning 180 degrees without moving your feet as you would do for a happo-giri? it seems very unwieldy to pick up each foot and put it back down again for this movement, unless you are jumping, which has its own problems.
Please don't tell me I will never get it because I don't understand aiki.