In post #74, Mathew offers one possible advantage to the Head Center Pivot -- you can tell if you moved your head because you can see under the ball more.
Is that the only advantage?
The Head Pivot Center is --ideally -- Centered between the Feet. In addition to the sensory advantage you have mentioned, it also offers an immediate visual confirmation of compliance with the Pivot Center requirement. It is an absolute, just as is the Straight Plane Line and the Flat Left Wrist and the Parallel to the Plane Line Clubshaft (when it is Parallel to the ground).
It also serves to Center other mission-critical Components -- the Shoulders, for example -- in preparation for their Uncompensated return to Impact.
Despite these advantages, the player can choose to locate that Head Center behind or in front of the Visual Center.