I think we are talking about different things. I mean, for me “public” doesn’t mean “federal”. For me means any governance unit, from local government (council), to regional (or state) to national. Those different units coordinate to offer that public health services.

My point is, public health services work and are an option that works well. There might be private alternatives, but those alternatives won’t incentivise government to also pass laws that improve general health care like the ones that ban smoking on public places or that limit the amount of sugar that is allowed on food (there is a nice essay about mcdonalds that explain how they use sugar to make you eat more).