At most a few countries worldwide has anything resembling "state controlled healthcare" (in fact, I cant name any - Norway used to a couple of decadds ago, but bow have plenty of private providers)

Almost all countries with universal healthcare have widespread private services.

Many have their universal services provisioned entirely or in part from private providers.

When I had a gastroscopy a while back it was done in a private hospital but arranged by the NHS and paid by the NHS.

If I want a private MRI there are dozens of providers in London.