Are you sure the US health care system delivers less care? I've read data about the health of the nation which seems bad compared to other countries (in relation to wealth). But I don't think that is automatically the fault of bad care. For example, if cheap access to food and cars make people fat, leading to bad health outcomes, it would not be the fault of the health care system.
It could be even worse, and a better health care system could make people be more careless about their health, because they would rely on being treated well enough to survive anyway.
Not saying any of that is the actual case, just that there are many aspects to consider.
As for controlling rent seeking behavior, wouldn't the insurance companies be interested in paying less? I don't see how private insurance facilitates uncontrolled rent seeking behavior?
It is very much a problem with public insurance, when patients never even see the bills.
In general, health insurance is a hard problem.