practical privatization of charity

My pattern is to be silent on politically-charged (in the USA) issues. One huge category of these is the overall role of government. What is the economically-optimal and/or morally-just ratio of the "public" and "private" sectors? Which achievements benefit from government's unique qualities?

Well-intentioned Christians differ on to what degree government should direct resources into charity. By "charity" I only mean supplying a cost-free individual benefit of some kind to recipients who can't afford the market price (1,2). Many Christians seem to think that government is an invaluable instrument for large-scale charity. Many other Christians seem to think that government involvement in such charity is too costly in terms of the reduction in private liberty (3).

I'm not interested here in producing arguments for either opinion. But what I find striking is a significant missing part of the debate. If someone supports a near-total privatization of charity, then doesn't that position entail a realistic plan for doing so? For government to drop charitable functions, it must be assumed that people will make up the difference. If people will be better givers without government in the way, then how will this occur? There's no question that less government giving can result in a lower tax rate, but afterward what percentage of those former tax amounts must be exclusively redirected to charity in order to continue to care for the poor?

This is a pivotal consideration for the plan to work. Assume (4) the plan subtracts from a particular person's effective tax rate percentage. How much of that difference will the person dedicate instead to direct private charity? Statistically, some people will end up giving less and some people more. So in the aggregate the question turns into the average amount that people contribute from their tax cut. When everything is summed, do the poor come out ahead or not after the proposed privatization of charity (5)? Furthermore, Christians should keep in mind that they'll likely need to give in excess to drag the average back up to a desirable range despite the "unprincipled deadbeats" who'll opt to take the tax cut difference and spend it on electronic geegaws, for example.

Public-sector charity has considerable administrative inefficiencies (6). Private-sector charity has considerable moral inefficiencies (i.e. personal selfishness). I think it's worth asking which is more detrimental in practice.

1. Obviously the definition of the Christian virtue of charity can be much broader.
2. The benefit could be money itself, in which case "can't afford the market price" refers to not having income-yielding capital/assets. For a typical worker who trades their labor for income, the corresponding capital/assets might be education/skills and the necessary physical and mental capability to do the job. 
3. I purposefully exclude from consideration a possible third group of "Christians" who flatly object to the mere monetary cost of charity.
4. I admit this would be more convincing with the actual numbers, but I don't have the information.  
5. Of course, this calculation should also include resulting differences in the poor's taxes. However, in the case of most poor, their taxes are very low both before and after the plan takes effect.
6. And so do many huge privately-owned foundations.