In Switzerland, health insurance may be private, but is mandatory and there is a large catalog that even the cheapest insurance must cover.
On the other hand, the state (each canton) pours large amounts of money into hospitals and research. There are laws and regulations at various levels (edit: eg how many doctors may open a new cabinet), and in these days we are reminded that doctors can be mandated to work when a crisis requires it.
Calling this a "completely private healthcare system" is a bit misleading.