Yeah we do. For many reasons:

Privacy is privacy. I ideally don’t want any of my data sold to anyone, but health data is even more vulnerable.

In my country it was even a big deal when they allowed different doctors to access your health data via a common system, as there were e.g. concerns that the information recorded by one doctor might bias another doctor, so some felt that it should be your choice what data to share between different parts of the public health system (except for explicit referrals).

Moreover, most European countries do have private doctors, private hospitals, and private health insurance – it’s just way less used than the public system. Those would have the same concerns as in the US.