As I've been taught, the foot pivot exists to facilitate the power being generated by the hips. Turning the foot itself mostly affects the range of motion for the hips and the ability of your muscles to work properly. The reason the foot has to turn is because for let's say a straight right the back foot has to be rooted and drive the movement while the left foot has to stabilize and help pull the left side of the body around. If the back foot doesn't turn the hips can't turn fully, and also the leg can't be in an optimal position to dig in without damaging the knee (try it, you'll feel it unless the foot is so far behind you have no weight on it). Turning the foot lets your glutes and other major muscles power the movement by pushing through that foot into the ground, which is what turns the hips.

So I agree with you that the power comes from the hips turning, and then to take that a step further the power from your hips turning comes from the muscles in your legs being in the best position to generate force in the way you need. In a way the power is coming ground up, but that power in the ground is really starting in the legs and being pushed through the feet then through the hips.