Word might get around about the boss, but when your job is on the line most people tow the line - as the Japanese say the nail that sticks out is the one that is hammered.
The problem with using metrics of what patients and parents think is this is not a good measure of what makes a good doctor or teacher. You can be an incompetent doctor with a fantastic bedside manner or a teacher all the parents relate to but who can't teach a fish to swim. The metric here is not the one you really want optimised.
On this topic I personally prefer doctors who have very poor beside manner because I assume they must have optimised competence over personal relations.
I think it's natural to assume this is not a good measure. Let's take teachers. I don't believe a teacher can consistently fool parents into believing they've taught the kid a lot when the kid is obviously not learning. And bluntly, the teacher is supposed to be working for the parent. Why shouldn't the parents decide who they want teaching their kids?