Just a "mea culpa" -- you're right, Medicaid and Medicare *are* a subsidy in the sense you meant: I thought you meant like paying people *not* to grow crops.
Cheers!
The farm program has some interesting parallels with what you're proposing for the health situation. In the farm program, first the government artificially drives up the price of food, through payments to farmers, import restrictions, direct purchases of commodities, and so on. Then, the government thinks food is too expensive, so it gives some of the people vouchers to pay for their food, and gives free food to others.
Another similar situation is the child-care industry. (This is particularly interesting from a health standpoint, because day-care centers and preschools are the #1 top infectious disease growth point in the country. If there were no daycare centers, there would probably be no significant infectious disease outbreaks.)
In this industry, the government directly pays some of the operating costs of some centers; it provides some customers with the funds to pay for day care; and it offers other users a tax credit for their costs. And then, having observed that families are struggling with high taxes, the government gives credits for children, whether they attend day care or not.
We have to remember that in all these situations, as in the health situation you're arguing, the money that's being distributed by the government is confiscated from the taxpayers under threat of armed force, based on criteria that have nothing to do with the issue at hand. "Would you kill your mother to pave I-95?" as P.J. O'Rourke famously asked. Or "Would you kill your mother so your brother could have free vitamins?"
My opinion on all this is that it would be better to remove the government's distortions from the market for all these various goods and services, and allow the citizens to make their own, uncoerced choices; pay for their choices themselves; and accept the good or bad consequences of their decisions.