It’s a mixed bag, but the funding source doesn’t necessarily make it a controlled market, to the degree that Medicare and Medicaid pay non-government providers and allow competition (which again, is mixed). Medicare and Medicaid coverage make up one third of the US population. The other two thirds are on group/employer insurance, private insurance, or no insurance at all.

For non-seniors, the medical insurance system certainly sometimes doesn’t feel like a free market from the consumer perspective, but the insurance companies are private for-profit institutions, and the medical providers are too, so it may well fit the definition.