You are making the mistake of conflating state-run with state-financed. Just because the state is financing universal health care doesn't mean it has to run it. As I mentioned earlier in this thread, only Spain of the developed nations has a system comparable with the UK.
Our neighbour European...
The only country that operates a similar system to the UK is Spain. Theirs is not an imitiation, it's actually a left-over remnant of the Franco dictatorship! Their spending on health is a little lower than the UK both per capita and as % of GDP. Spain manages better outcomes generally though.