That's the same model we have in Canada here mainly. Provinces pay for the service, but family doctors and most other practitioners are private businesses that then bill the government based on a set fee schedule.
The exception I guess is hospitals which are kind of a mixed structure of some kind and doctors in them are salaried employees not businesses of their own.
The problem with this approach is it intrinsically creates conflict between the doctors and the gov't about how fees are structured, and the patient gets stuck in the middle. That and inconsistent standards, structure, and quality. Couple that with conservative governments that sometimes have ideological antipathy to socialized medicine generally, and there's a recipe for difficulties.
I'm given to understand that the NHS in the UK is not like this, and most doctors are salaried?
NHS services originally (tories have been chipping at it a lot) had all NHS services delivered by NHS staff and infrastructure, with I think one major exception being dentistry.
In Poland we have public insurance system (also slowly killed by government) but in reality nearly all providers are private or, at most, owned by local government (city etc.)
Apparently South Korea is doing pretty well with its healthcare: financing is public, service delivery is private (but heavily regulated).