Most countries, including most in Europe, dont rely on a state run health care provider to provide the actual health care. The state in these countries often provides (sometimes partial) funding for the citizen to get their own treatment from private providers.... A bit like schooling with school vouchers.
Do these private providers restrict access to those on the lowest incomes by making them jump through pointless hoops? No. What's your point?
The evidence suggests that services provided by the state frequently actually provide rather poor value for money.
By building an infrastructure for such a scheme your promote competition between providers that may assist in driving down the end user costs and /or improving the service provided.
As above your argument is the one that leads to inferior and expensive services.
All great theory, practice tends to see private companies making government projects go wildly over budget and delivered very late. Or in the case of healthcare, people get saddled with crippling debt for fairly basic medication.