It depends on the level of market failure, but there are not a ton of hospitals to choose from regardless.
https://fortune.com/2025/11/10/nvidia-hometown-santa-clara-c...
Monopolies, in these cases natural monopolies, can in fact exist. Look at the Micro supply and demand curves. As a general rule over those 100 years, there has not been rationing of electricity. There are natural blackouts and today an unplanned surge in demand (as happens in every industry such as chips after Covid), but generally the price regulation did not cause some kind of gas lines.
Whether they would be the rate limiting factor in health care remain to be seen, since health care is highly regulated with regulatory capture, licensing, and violence enforced market manipulations. As a thought experiment, in the extreme that health care were a pure monopoly, then I could envision some price caps somewhere between cost and price where the supply curve is relatively flat on either side thus creating minimal effects to supply.
Typically in these countries you actually can get health care as long as you pay privately yourself and don't go through the 'single payer.' A price cap would mean that no matter how much you're willing to pay, you can't pay over the cap, which is much rarer than the presence of 'national healthcare systems' that merely won't pay over the supposed soft 'cap'.