E.g. in the UK 10% have private insurance (though very often as a employee benefit that they don't pay much for and so the takeup is artificially inflated that way).
But presumably the fact that they have to convince people it's worth paying extra on top of their taxes provides massive additional competitive pressure and is part of what helps keeping the prices low relative to the service provided.
E.g. I've seen cases where it'd pay for Americans to fly to London and stay in a luxury hotel and get treated at top private clinics in London vs. getting relatively basic private treatment in the US.
Note that private insurance in the UK does not cover chronic conditions - they shunt you off to the NHS for anything that isn't easily curable. That's a large reason why it's cheap.
