Right, and in my country you can even mix and match it.

I went to see my GP, paid for by public health, they referred me to a specialist.

I chose to pay €100 to see a private doctor who was available sooner (the next day) and had better ratings.

They referred me for an MRI which was done at another private provider, paid for by public health.

I went back to the private doctor and paid for a non-surgical treatment, which wasn't available on public health.

If that doesn't work, later I can opt for surgery, paid for by public health.

And even more importantly: There is one system that tracks all diagnoses, treatment, medication etc used by both public and private healthcare providers, so medical history is available instantly to everyone.