The dynamics of the financial incentives involved are a bit different with goverment provided health care and private clinics funded through medical insurance.

Private clinics which are the main receivers of insurance money are often owned by the doctors. Thus, they have an incentive to over-charge, over diagnose and run expensive labs whenever they can. I'm not saying it happens, just that the incentive is there. Government provided healthcare offers no financial incentives for practicioners to over-diagnose - on the contrary this system then incentivizes to be as prudish as possible with the public resources.

I'm not saying one or the other is better or worse, just that they are not equal in every way.