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.
Regarding prices, yes insurance companies do negotiate prices to some extent but that has not controlled prices. See $20 asprin tablets given at many hospitals. There's a ton of reasons our private system has not done better controlling prices. It's hard to shop care. Insurers pass service costs onto employers who pay premiums who aren't consuming the care (the employee is). Medical companies find weird workarounds see medication couponing. I could go on. What it amounts to in the end is exactly rent seeking behavior, whatever the free market should be doing it is not. And there is no clear data that it is functioning more efficiently than a public system or at least a regulated market like Switzerland has.
In the end there's a reason that every developed country in the world other than America controls healthcare prices the same way. The government or a body backed by the government negotiates prices effectively fixing them.
Of course health insurance is a hard problem. No one ever said it wasn't.
This means a greater share of GDP goes to pay for worse health outcomes. The cost of that lost GDP is measure in opportunity costs. That is GDP that could go to education, infrastructure, social programs or simply more savings for the median citizen.