Most USAers assume that the welfare state is more needed and more effective than it really is, and that a huge amount of charity (in fact more than we could ever come close to getting) would be needed to replace it. They may be right that charity would never be enough but I think charity would be much closer to being sufficient than people think.
When progressives assume that the those lowest income cannot affords something they often forget to factor in reduced demand lowering the price.
One example is that in a market for housing with restrictions on building (adding housing units) which we have in most counties, and where housing is subsidized, if you remove the subsidy the landlords will take some of the hit probably most of the hit.
Another example is that as far as I can tell, even adjusted for inflation, our USA schools spent 3x as much as in the 1960's and output is not much improved. Now I am sure that the experience for the student is better now but I would bet that you could cut spending hugely and get pretty much the same output. And if parents payed directly you could possibly get better output due to motivation and for some other plausible reasons. See the link in Scott's post on this.
Welfare also seems to not make as much difference compared to what most people would expect. Take Social Security, our biggest welfare program, we only need it for low savers who are to weak to work who have not family or friends to care for them. That is a small percent of the population but to reach that small percent of the population without creating very bad incentives we have to give SS to everyone. A huge cost relative to the problem. BTW I think we should give every USA citizen over 67 $200/week from SS, reducing the program cost by about 1/3rd. BTW I think absent SS most people would start to save as much as possible at about 60 years old and work as long as they could and do OK.
Also I once saw a show on PBS about a guy who built a road to Miami Beach.
Narrator: One of his greatest ambitions was to persuade the government to start building better roads. He convinced the leaders of the automotive industry to jump-start the process by financing the first paved road across the country -- the Lincoln Highway, from New York to San Francisco. It was such a success that he went on to build another road -- the Dixie Highway... leading, very conveniently, from Indianapolis to the foot of the bridge he was paying for back in Florida.