With single payer, taxes cover the cost of care for everyone. With private insurance, the costs for more expensive care is covered by the premiums of healthier individuals, which is sort of like what you suggest but does not financially punish people who get sick.
What you propose may not be workable (or humane, for that matter) because it distributes the costs of extraordinary care to people who already have more-than-routine care. Presumably these people already have paid their premiums and now would shoulder most of the cost for maladies such as broken bones.
EDIT: add missing word "workable".
The main premise of this discussion is that it's not possible to pay enough for healthcare. I.e. even if you taxed people 100% there still would be things they couldn't get because it has to be available to everyone and there are always new treatments being developed, new machines, new drugs.
Whether you find it humane or not is based on what you consider the end-game of healthcare.
I am not sure where you get that I want less people to go to the doctor. I just want to be able to pay for the situations you can't afford by asking for money in situations where most people can. And yes if you don't have any money then you should get help, but for people who make enough it should cost something.
So there is nothing paradoxial going on here. You just assumed I wanted fewer people to go to the doctor which I never said anything about.