1. Any significant gaps in coverage pretty much screws up the whole thing
2. Relying on people's good nature is never going to work well in a large and complex society. This sort of thing works great in a small town, but it always depends on relationships because that is how people function.
3. These sorts of things tend to need large scale coordination
When I think about the idea of charity handling this sort of thing, I am reminded of Haiti. When the earth quake hit, the red cross was flooded with cash and a few things happened.
1. People got pissed that the red cross did not spend all of it immediately and tried to hold money back for long term concerns (this hits points 1 and 3 above in terms of issues.) Funding to the red cross dried up quickly because haiti is another country, out of sight, out of mind (point 2). Because there was no real coordination (the red cross only had resources to do so much) the people in Haiti are still loving in poverty camps, having huge problems with water born illnesses, crime, lawlessness and other typical problems of poverty. One of the major issues there is that land deed records were somewhat destroyed during the earth quake so nobody knows who owns what, so the whole private property angle, pulling yourself up by your boot straps idea didn't pan out either.
A similar thing happened last year when several places were hit by tornados in the south and near Joplin. There is a section of the town I live in right now where you can still see tornado paths with ruined houses and what not from this time period. People forgot about the problem.
Ultimately, this is a problem of the general public vs experts. Experts tend to be able to keep focus over the long term and can be protected by an entity such as a government to do actual long term good rather than what happened in haiti.
This is what I see as some of the major differences between private and public approaches to issues like this.