That’s a blanket statement that leaves too much wiggle-room. The UK’s NHS, for example, is partly private insofar as GPs and many other doctors work for themselves and contract themselves out to the NHS instead of being NHS employees. Correct me if I’m wrong, but a group of doctors could form an organization that forces its patients into binding arbitration for disputes while still being contracted to the NHS. It would result in a backlash of public opinion - but still possible.

The NHS also has other private components: it used to run its own logistics division which was scrapped in recent years and replaced with services provided by couriers like DHL.