It would be difficult, and fiendishly expensive, to unpick the NHS universe while simultaneously constructing the network that underpins most of the European systems, not least because there's no single employer of last resort like the NHS. Of course, lots of routine services are contracted out, but the NHS is one of the two or three largest employers in the world.
Most GPs are sole practitioners without nursing support (because there are community nursing hubs are private businesses/practices. You make an appointment separately for routine infant care, wound dressings or vaccinations and pay separately).
Labs are privately owned too, and the patient drops off the samples and receives the results personally rather than via the doctor.
Big city and teaching hospitals may be state operations, but many of the smaller ones are owned and run privately or by charities. And France is still short of GPs, especially in rural areas.
The cost of healthcare is an issue everywhere, because of demographics and also because of medical and pharmaceutical advances.
Although it would be lovely, the transition would be brutal.