Hmmm, totally charity-funded system.
Never generates sufficient cash for this sort of need. People who would pay a small amount in a tax-based system, pay nothing because they are strapped for cash and see it as the responsibility of the rich people. Rich people think they're some sort of bloody saint if they make a big, flashy donation, get their name engraved on the neonatal cot or whatever it is, but in fact the amount of the donation still might be less than they'd have paid on their large income if it had been taxed.
People choose which healthcare charity they donate to, and they want their money to go to the neonatal cots or the MRI scanner, and geriatric psychiatry ends up with nothing.
Healthcare competes with other charities, so one day a lot of people decide that they're going to give the money to the Somalia appeal, or the Pakistan earthquake appeal, or maybe the dog and cat home.
And there are people who won't donate at all. In any system. If you allow them to opt out, they will. So everybody else's contributions have to be higher to make up for that.
The only vague possibility that I could see would be to send people an optional tax bill. You would be told the total amount of tax the government thinks you should pay, all broken down and hypothecated to different purposes. Healthcare, education, policing, defence and so on. You would tick the boxes of the things you wanted to contribute to, and send a cheque for only that proportion (which will of course be higher than in a compulsory system, as the government will have figured out what proportion of the population won't pay, and hiked the nominal amount accordingly). And there would be no comeback if you didn't pay anything.
Sounds great. I wonder why no country actually follows this system?
Rolfe.