Right on!
That's exactly right. Your hips turn and, if your legs remain springy, your feet will get dragged behind as much as they need to. This also prevents you from opening your front foot more than necessary and exposing your shin and groin, which beginners will often do if they think about initiating the turn with the feet.
With regard to the pivot point being the heel or the middle foot:
From my perspective, balance is most critical is when there is a clash between me and my opponent. The resulting force between us will invariably drive my weight back toward my heels. The pivot is used to allow that force (in that instant) to go by me, to bypass the force. it is the force from my opponent that drives the pivot.
All this to say that I want to be very comfortable turning on my heels because that is where the weight is going to be when I really need to turn. Hence, I do chum kiu with heel pivots.
Once, my sifu told us to train with our weight right at the back of our heels where it felt like we were about to tip over. I thought that was very bizarre until I understood the exercise: Don't panic, just absorb by sinking and turning.