I can’t speak for all EU countries, I’ve only spent considerable time / have experience in 6 or 7. Where I live (in Austria, for the last 25 years) there is considerable preventative care included in the “Medicare for All” type system here. For example, every year you can get a physical exam including blood work. Age relevant tests are free as well. This includes things like referrals to get an x-ray, a mammogram or a prostate check.

It makes no sense to privatize the profits from preventative care while socializing the upfront costs. The health of a nation is important to each society (nation). It not only makes no sense to privatize intrinsically societal needs like the healthcare, armed forces, the fire department, the police, schools, sanitation, etc., it is actually against our long term interest as a society. We used to have private fire departments and private police. It is long since time we moved on from private health insurance (if not private healthcare).