As a general rule, insurance companies do not control the costs incurred by producers whom they pay. Hurricane insurance companies do not control the costs of re-building when a hurricane strikes. If the clients buy insurance for replacing their structures, then the companies agree to pay the costs of replacement.
With health “insurance,” the companies agree to pay for medical procedures. They cannot then “control” the costs of providing the procedure. If the state is subsidizing the companies, then they have more money to spend to pay for the procedures and the price is bid up. If one company tries to keep it’s costs down by paying less or substituting a different less-costly procedure, then, other companies will move in to earn the higher profit. To get business, they will have to provide better terms. That means they spend more and healthcare prices rise.
This is one alleged justification for single-payer system. If there was just one “insurance” company, then the competitive process could be eliminated and healthcare prices held down. I’m not saying I agree with this assessment, only that some people argue this way.
On why the American healthcare system is so much more expensive than those of other countries, it seems to me that one has to entertain the possibility that the system is designed to be so. It has been, after all, constructed by doctors. What is cost for patients is income for doctors. So we have a system designed largely by doctors which makes doctors rich. When insurance companies try to reduce costs by offering limited policies, the state makes doing so illegal, as we’ve seen in the roll out of Obamacare. The state mandates that insurance companies cover all sorts of things because it means that more money will be spent on doctor’s services and other beneficiaries of the system.