They DO have a very precise concept of what they want in a health system: that they do not want to have to pay for it.

These are greedy, acquisitive people. They don't see health care as a social public health issue, like garbage collection, sewage treatment, or water purification. They see it as a handout they have to provide for lazy stupid people who can't afford their own doctor and medicine. And like all greedy, acquisitive people, they are terrified that someone, somewhere might have a hand in their pocket; a right that they reserve solely for themselves.

This attitude is similar to the one they have for education. They don't want education to be free, paid for publicly, because then anyone who qualifies for education can receive it; which would mean they would have to compete economically with an educated populace.
They want university education to be just expensive enough so they can afford it, but no one else. Of course, this does not mean they are willing to pay fully for their own schooling, they want the state to subsidize the bulk of it, as long as only those who can afford the price of admission get to benefit from that subsidy. And its why they are so fanatical about "local control" of public schools. They reserve the right to pay top dollar for schools for their kids, but they have no intention of paying anything to educate anyone elses.

If they could find a way to only pay for THEIR police, courts, or mail delivery, they would, and to a certain extent, this is what private security guards, expensive lawyers and parcel delivery companies are all about. Meanwhile, the public institutions that provide these services are starved of funding and gradually replaced by their for-profit counterparts. Hell, even prisons are being privatized and convicts farmed out as cheap labor. In fact, the insurance industry is another mechanism to do this--reserve basic social services only for those who can afford the price of admission. Socialism for the rich, free enterprise for everybody else.

The health care debate isn't just about health, it permeates every level of social organization and every infrastructure institution in this country. The wealthy don't want to be taxed to pay for anyone's civic services but their own, but they have no objection to taxing anyone else to help pay for theirs. It's just that in health care, the consequences are more obvious and more deadly---and much more expensive.