GG,
A few notes. I think we do agree that gov't programs are run very inefficiently and often are ineffective. Overhead costs, scams, nobody taking care of business... I'd wager only 1/3rd of taxes, at most 1/2 of taxes set aside for welfare end up helping people who really need it. You can adjust that number up or down, but I think that's the real range.
Yes, there are plenty of people who donate nothing to help the needy. That's ok, we do not need 100% participation. We didn't have it 100 years ago, we don't need it now. When you said 'public hands', I assume you mean non-gov't operation of charity?
If so, there are lots of ways. First, charities have very little overhead compared to gov't programs. People who work/volunteer with charities really care, and will mind the store. That will maximize the benefit to people who really need it and block out many who are just scamming. Not perfect, but far better than gov't programs. Charities rarely give out cash, and instead provide food and clothing and shelter to those in need. Again, this filters out huge deficiencies in gov't programs where people 'buy' (with taxpayer money) all kinds of things. Or how many people sell their food stamp money (yes, this is a $1billion scam if not more annually).
Look back. There was almost zero welfare during the great depression, yet hardly anybody starved. Hungry? Yup, and lots. But people helped one another and as a nation we got through it. An example, after church we had lunch with a couple that regularly donate to help the elderly/needy with their utility bill. They don't advertise it, they know someone at the utility company and that person pings them when there is a charity case where someone is about to get their utilities turned off due to non-payment. They give the money to bring that person up-to-date. No fan-fare, no scams, no gov't involvement. Simply good people blessing others at the local level.