What happens in practice vis a vis National Health systems and the Insurance companies is that the private sector offers plans that supplement the services that National Health does not offer. The cost has to be competitive with the Government services otherwise people will not subscribe. In Israel, once the HMOs that provided medical services were taken over by the government, they "added" features that were not covered. Complimentary Alternative Medicine, Dental Care, and other services are offered at a rate that is within the budget of most people.
Often enough private services, and providers have contracts with National Health so that they offer discounts to the public, in order to "increase their volume"--does it matter where the money originates? It becomes a matter of "enlightened self-interest" to charge less than what they would ordinarily take and get more "customers".
I am sure that there is a similar pattern in the EU, and the insurance companies, and private servers don't seem to mind. What is interesting that some medical procedures in National Health actually benefit the doctors, the ones that may not be as critical that per force bring extra income for them.
In any case, a National Health system that is based on Single Payers works big time, and the silly questions that people ask the quality of the service disappear as soon people "get used to it". Zev Davis