I think you are missing the point of the system: you pay the difference in general income tax now so when you are unable to pay such levels the state (and those still paying income tax) take care of you. If you are always able to pay for private health insurance or are always being taxed highly on income you are very lucky and should have no problem kicking in for public health.

There is no minimum tax payable before accessing healthcare services: if you have no income, or are unlucky, you won’t die or be debt ridden for the rest of your life.