Exactly right.
There are several ways to deal with medical care costs.
First & foremost, level the playing field. A law which makes it illegal for any care provider or medical facility to charge two different patients different prices for the same procedure, but that cash discounts were allowed, would do wonders. I have insurance, and am always amazed when I get a statement showing that the Dr. charged $180 for an office visit, but that the negotiated rate was $16.83 for that and he has waived the remaining amount. If I could pay the $16.83 I wouldn't want or need insurance in the first place. But if I go in without insurance I pay $180.
Not only is the price lower, but the Doc has to hire a special employee just to file the fool insurance claims. If we eliminated her I bet he could make money at $12 for the office visit.
Secondly break the AMA stranglehold on the field. Let me choose to go to a full-fledged Dr. with a degree and a diploma, just a first aid provider for a lot lower price, or something in between. A first aid provider would be able to tell me if I needed more than he could do.
Third, reform the rules for the health insurance industry. Right now, they have an incentive to deny every claim and then only pay after it has been re-submitted. This is a huge waste of manpower. They pay people salaries to deny the claims and then end up paying them anyway. The salaries are wasted.
You hit the problem right on the head -- maybe without even meaning to do it.
The only reason you have the sort of idiocy you describe is that health care has become constructed as an entire sector of the economy where people don't pay for products and services directly. You'd never hear of such an outlandish scenario where you walked into a Verizon store and bought a new iPhone, only to have all the negotiations conducted through a third party that would "deny claims" and "negotiate prices" until your $600 iPhone only cost you $76.28 ... but that's only because you've been paying "iPhone premiums" of $20 every month for the last ten years.
Anyone who thinks this kind of business arrangement would result in (1) better iPhones and/or (2) more affordable iPhones would be laughed out of the room.