The US system is neither fully private nor are those other systems fully public. Its a false binary people use so they don't have to actually engage in the details of how different systems operate.
Nobody is arguing that the US has a good or 'the right' system. And nobody should argue that the British NHS (or systems like it) is some amazing magical place that gets everything right.
> When will the American people wake up and realize their private healthcare model only benefits the rich that are able to afford the care.
And yet one of the countries wit the lowest spending % of GDP on healthcare with very good outcomes has a system that has a lot of private aspects.