The UK has a mixture of private and public healthcare.  You can wait to be treated on the NHS or be treated immediately if you pay.  Same doctor, same hospital.  If you pay you go to the front of the line.

The results have been catastrophic and the NHS is close to collapse.  A third of the junior doctors quit every year.  Clinics are full and you may have to travel great distances to find one accepting patients.

Economists love their theories but in many cases they do not work in the real world.