It is beyond question that the healthcare market in the US is terrible, because its expensive and ineffective. That is out of discussion for any reasonable debate.
What is also out of debate are many reasons why is it so expensive today: restrictive immigration of doctors, importation restrictions on drugs, self imposed gov restrictions on bargaining, hospital building regulations, malpractice legal costs.
Government taking full control of the system fixes NONE of the problems listed above, which are by and large some of the biggest price drivers. It might make them worse, because all of the ones above are exclusively decisions of the government.
> you can't just apply the exact same per capita spending...
I agree, tho at the same time you cant expect the government to be more efficient as it increases its scope of scare. Economies of scale dont work for the government. 10x service might cost 50x. It should be clear however that its a reasonable order of magnitude: more than barely double digits. A 20% gdp cost on healthcare will come with cuts in service, I guarantee that.