The US occupies the large gap between "wholly run publicly" and "the whole thing is private". Besides public-payer insurance, there are government hospitals in the US too.
at teh government owned hospitals, do people still use "insurance"? or is it free at the point of use? what criteria applies to treatment there, limitations etc?
It all depends on which program/hospital/patient/government we're talking about. Everything about government in the US is fragmented.
Scenarios range from completely free at the point of use, to 100% out of pocket, and everything in between.
The 5 programs I mentioned above are just the largest federal programs. States and localities in the US often implement their own programs. There are hundreds of government healthcare programs that someone in the US might take advantage of, depending on where they're at.
The problem with healthcare in the US is not that there is no government healthcare initiatives. The problem is that we have too many. Fragmentation is inefficient and expensive.
Governments in the US spend $4,854/person/year on healthcare. The NHS spends $4,192/person/year.
The US has several public insurers.
About 72 million Americans are enrolled in Medicaid or CHIP.
About 60 million are enrolled in Medicare.
The MHS covers about 9.5 million people.
The IHS covers another 2.2 million people.