... the problem with the argument is that people are concerned about the free rider problem. There are certain burdens that people think that should be shared by everyone, in some sort of fair manner proportional to ability to pay. People are typically unwilling to donate their resources if they think that others aren't going to do so. And even if you think that 99% of the population will contribute, your contribution will make so little difference that you have no incentive to contribute.
If the free market is a game where the object is to accumulate or maintain wealth, then you have no incentive to part with your resources. The only thing that might compel you to do so is compassion, but as long as there are other compassionate people out there, then you might figure that they can take care of the problem. On the other hand, a "public" charity, and the same rational applies to other public burdens like police and national defense, gives people reassurance that everyone is sharing in the burden. Nobody gets a free ride. Nobody gets filthy rich because all his neighbors are compassionate and he isn't.
In other words, people are willing to vote for public charity that will cost them money provided that they think it is fair.
As a practical matter, if you have a society that depends totally on voluntary contributions, then nobody really knows how much to contribute, because nobody knows how much anyone else is contributing. If only a third of the people contribute, then they would all have to donate 300% of the amount that they would need to contribute if everyone did. Individual contributors do not know how much to budget for charity, but a central authority would.
On the other hand, the problem with any taxpayer or third party payer payer program is that it becomes an all you can eat buffet. There is no incentive for people to not to take a hand out. The number of people dependent upon public charity tend to grow over time as more and more people become dependent upon public charity. To combat this, you need to have incentives built into the system.