I had private health insurance in Germany. It was quite expensive and had a very high deductible — so pretty bad incentives around routine health care, I never made a claim in 13 years.
The upside was that if you needed, say, a brain transplant for ten million Euros, as long as it was medically necessary they would pay for it.
Now I have a policy elsewhere that is cheaper, still covers me when I go to Europe, and has a much better copay structure while being 100% private. Downside is I can’t afford that brain transplant, but I’ll probably be OK for everything else.
But if a procedure is medically necessary in Germany, public insurance pays for it too (and if they deny, you can, often successfully, appeal or sue them). The biggest difference is that you’ll get an appointment much faster due to the quota system for public insurance patients, and that private insurance can cover more things that aren’t strictly necessary.