Health care in the US is rationed, all of it. The poor do not get all they want of every kind they want, and neither do the military.
The insurance companies quantify the risk by limiting their coverage and rationing the delivery.
If you are wealthy enough to pay for it, you can get medical coverage that is essentially limited only by your own inconvenience in accepting the attentions. But very, very few people on this planet can afford that.
The vet business is actually pretty tightly regulated. The lower prices are not from lack of regulation, but from market economics - so we have an example of free market health care that lowers prices. And so we can see how a market in health care would work - what the conditions necessary for a free market look like in the health care business. They include, fundamentally, not buying the service.
As far as forcing everyone to buy insurance, without doing anything about the insurance companies' denial of care or rate adjustment options, a better guarantee of continuing the current arms race inflation of insurance premiums and profits of insurance companies is hard to imagine - and the effects on prices easy to imagine.