It's neither capitalist nor socialist. If it were fully socialist you would have long wait lists but it would be free and there would be one payer. It is like this for Medicare and Medicaid which I've heard are a fantastic UX. But this is only the case for about half of Americans (the ones who don't pay for it).
On the other side, if it were fully capitalist you would be able to see the price and walk away if you didn't like it. This is what makes capitalism work. Your margin is my opportunity. Instead, the upper middle class, who pays for everything already, and is unable to use Medicaid, is forced to use a certain "network" of providers and never, ever sees the price upfront. This is the cornerstone of capitalism. Does the buyer like the price? If so, transact. It's completely not there. Instead, it's actively discouraged and banned, and the price is maximized post-hoc by the same entities who negotiate directly with the employee's employer. Ie, a quantitative shakedown.