I am sorry but this is simply wrong. There are social health and pension systems but both of them suck and are horrible broken. If you are under 30 and not nearly dead you should do everything possible to get out of those and take care of your self with private insurances.
It is broken by German / first-worlds standards, of course, with people not having access to something similar laughing about these claims, if it was remotely funny.
The other problem is that the prices are on a steady rise - the only easy way for private insurances to raise profits is taking it from their clients.
The way to go is public insurance with additional private extensions - it ensures that you can cancel your (sub)-contract without fearing about your general healcare plan.
There are ways - one example: just make sure you are employed and do not earn more than (roughly) 4500€ / m and tada: back.
"The other problem is that the prices are on a steady rise"
The same is true for the public insurance, beside they do not pay for the same services, medicaments,... as the private insurances. There are also better ways to prevent paying >600€/m when you are over 60. Just a matter of the right company / contract.
"The way to go is public insurance with additional private extensions - it ensures that you can cancel your (sub)-contract without fearing about your general healcare plan."
100% agree on this. If you are an employee.
//edit: not sure if this discussion fits here or if we, if there would be any interest, should use IRC or something else.