Sure, but I think the distinction lies in if toll booths (hospitals) are still private, meaning carte blanche payment from government money to private actors.

Carte blanche only if there is no ability for government to negotiate prices, or set up hospitals themself. There is also of course the protectionist bias of government licensing of doctors. (Doctors from the EU, Canada, Australia, Scandinavia,etc. aren't able to practice in USA, no matter how qualified, unless they undergo residency program in US, which have strict quotas).