It seems to me that should then be handled by the legal system or perhaps the licensing boards. That is to say, the insurance companies should be forbidden from denying coverage and if they believe that something is going on then they can alert the police or take the doctor to court.
That's simply unrealistic. Medicine is more of an art than a science and most treatment plans fall into gray areas. While insurers do file formal complaints with authorities when there is clear evidence of fraud (like a single provider billing for more than 24 hours of patient encounters in a single day), there is an enormous amount of waste and abuse by providers which can't be proven in court as fraud. The fee-for-service financial model incentivizes over treatment.
The fundamental problem here is that demand for healthcare is virtually infinite while supply is limited. Healthcare already consumes about 20% of the US economy and there is no excess capacity to expand it further without dragging everything else down. We can make some things more efficient and reduce administrative burdens but ultimately we have to draw the line somewhere and ration care. That's going to be true regardless of whether we have commercial insurers or socialized healthcare.
Most doctors are not very rich at all, upper middle class. I make more than many doctors I know as a software engineer, and this doesn't even take into account their debt.
Care is expensive because of insurance. Hospitals needs hundreds of administrators and billing specialists. Very little of that money goes to the doctor.
Doctors deserve to be rich. They do incredibly specialized work where the outcomes are extremely critical, to put it mildly. I don't want doctors performing unnecessary treatments to line their pockets, but I don't think we're actually in danger of that.
The difference between a rich doctor and a rich insurance exec is that the doctor helps people, and the insurance exec makes their money withholding necessary care from people, in effect torturing and killing them.
This is the problem with insurance in general. The incentive of insurance is at direct odds with patients and doctors.
It's a tug of war, and even if insurance "loses", it doesn't matter, because you lost trillions of dollars in that tug of war. It's artificial inefficiency caused by privatization, very much a Dark Night burning a mountain of money situation.