Only if they want what the doctor has!
Say I wanted your house or car. I mean, what do you expect, that you should be able to charge whatever you want and I should have to pay it?
Mind boggling indeed.
Buying a house is not absolutely essential to me. I can stay with friends or family if need be, or just rent an apartment.
Buying a car is not absolutely essential to me. I can hitch a ride with a co-worker, or walk, or catch a bus, etc.
But if I'm an insulin dependent diabetic, healthcare is pretty essential to me. Or if I have cancer, healthcare is pretty essential to me.
So that analogy doesn't hold much water.
According to your logic, by being able to charge people whatever they want, doctors would have people at their mercy. If they could charge whatever they wanted, our healthcare system, economy and national health as a whole would be even worse than they are.
In a perfect world, I would have no problem with doctors setting their own fees because--ideally--free market economics would take over, and the doctors who charged the most reasonable, affordable price would succeed, and the doctors who charged ridiculous prices would either go out of business or bring their prices down to compete with the other doctors.
But that's not happening in the real world. We haven't yet made it to Libertarian-Utopian-Nirvana-Land yet. In reality, healthcare in the U.S. and all over the world for that matter, is dominated by insurance--both private and government-run. Insurance companies are pretty good at turning profits. They generally won't pay for something unless they have to, and even then, they decide the price they will pay. Insurance companies, love them or hate them, (And I hate them) are the great equalizer. They prevent doctors from charging people outrageous prices for essential healthcare services. Without insurance companies or government regulations, unscrupulous doctors --in collusion with each other--could form a sort of monopoly, and charge whatever they wanted. I guess that would make some people happy.