Thanks for this overview of the underlying issues!

As a point of reference, Sweden used to have full government funding and government provision of services, except for dental care where we had private providers.

These days, all types of services follow the dental care model: The patient can choose where to go, and the government "insurance" pays for the services. It's called an insurance, but is paid through the employment tax and the premium is determined entirely by the salary.

You can also add a private health care insurance, and get access to further clinics and services, shorter queues, etc.

Counties license providers, so while in some sense anybody qualified to provide services can do so, each county may uphold e.g. a certain quota of private vs public providers.