Universal healthcare is superior to most “free market” based implementations because of economies of scale.
In simple terms, countries with uni healthcare use huge negiotiations to push down the costs of a treatment, so, let’s take a rather iconic example of the NHS, they advertise that the contract to supply them with hip replacements is up for grabs, and the company that can provide millions of their elderly citizens with said hip replacement treatment (this is overly simplistic but it explains the point) at the cheapest price following a minimum standard gets the contract. This minimum standard usually actually being quite high. The company who wins this contract basically wins a guaranteed source of easy income for years, that will be consistent and constant, a quite literal offer they can’t refuse, and so the discounts the nhs gets are often enourmous because competition for this contract is huge.
This is a big reason why drug prices are much lower, and why european countries and canada have often superior or similiar standards to healthcare to the USA while paying far less and even being poorer then the USA. The bigger the country the more effective this type of arrangement becomes. Universal healthcare would also remove the charge master problem the USA has, which is a key factor for high healthcare costs (an unintended and unforseen consequence of the free market).
The charge master of a hospital is basically the log sheet where they keep prices for treatments in a US hospital, and how much to charge for them. The prices are usually over-inflated. This was actually because of health insurance in the USA, who applied the same collective bargaining tactic to try to get cheaper prices for the people who they cover. Individual hospitals could not afford these discounts and actually had to raise the prices via their charge master to appease insurance companies, so essentially hike the price by say 50% and then the insurance company gets a " 50%" discount on that. Hospitals then later abused this too for their own gain and is a key factor in US healthcare prices.
Edit: This problem with healthcare prices by the way, was the product of private insurance, and was a serious issue prior to obama care in case you were wondering, although Obama care was in many ways a giveaway to said insurance companies.