But think about how the image would look if locked to the centre of the subject's face. Every little head tilt would translate into wobble of everything in the background. A simple slight nodding of the head would look very unnatural. Our eyes compensate for this, but on screen it might be very strange.

Intel has already solved all this and is about to release a camera that does it (see my sister comment for details). It is still a brilliant idea even if it is not original :)

You'd want to apply a steadicam-style effect to it, only adjusting if the face moves significantly away from that focus point, and even then doing so gradually.