A charity only model would not be anywhere near as effective as a primarily government based model. As a general rule any policy rooted on aphorisms like "People would be less fat if they ate less" or "People would give more to charity if they were taxed less" are doomed to fail. If you want evidence of this just look at abstinence-centric sexual education policies and Colorado. Birth control was made more available and affordable and unwanted pregnancies (measured in resultant abortions) went down. Better outcomes, most on both sides of the abortion minefield are happy and it cost everyone - including the taxpayer - less money overall.
Part of the problem is that people consider what the government does in these social programs to be identical to charity. This is absolutely not the case. I'm not really sure how this can be confused but the government does not simply hand out sweaters and groceries on the street to poor people the same way Goodwill, your local food pantry or church do. Charity is highly targeted. It needs to be. Private charity organizations do not have the scope or funding to implement equitable, broad programs across the entire nation minus a few exceptions like the Gates foundation.
As a case study let us examine NGOs involved with foreign aid in other countries. These NGOs always go for stronger public sector implementation because of two major reasons: 1) Scope; you can reach more people and 2) Rights; everyone has the right to a publically owned service while your rights to private services can be taken away. The issue of rights is already a problem that targeted charity can face that government welfare simply doesn't have to ( or, at least, if the issue arises in government welfare programs it is far less intractable than when it does in private charity). For example, Catholic Church provided AIDS relief is often underscored with a prohibition of condom and birth control distribution and a requirement of abstinence based prevention methods. This makes this relief far less effective unless another government or organization can provide sufficient pressure to override these conditions.
Then there is aid saturation. In exceedingly poor countries governments have no ability to coordinate a concentrated effort among the 100s of NGOs present in order to provide more effective relief. A large number of aid organizations present in one area makes it very difficult to coordinate and leads to waste, redundancy, or an over concentration of resources that threatens to leave other, equally serious issues unaddressed. Rwanda has instituted a national government plan for how to use their foreign aid resources and ever since this institution the number of NGOs in Rwanda has gone down considerably but the cost effectiveness of Foreign Aid has increased dramatically.
Let us say that we can somehow afford to institute a "national charity plan" in this country without levying any sort of tax on the people to pay for it (which means we pay for it using borrowed money). Having a third party arbitrate this plan is nonsensical as: 1) a foreign body managing the finances of the richest country in the world because no one in that country is willing to do it themselves is just out of the question and absurd and 2) no one understands the problems of a country better than the people actually running that country and most importantly 3) someone with real authority needs to be able to do this or you just risk unnecessary conflict between private and public charity sectors.
But let us go on with the hypothetical, we manage to draft up a national charity plan to use the people's freely given donations in the best way possible. The charity plan will then be subject to two major constraints: 1) the unequal and unreliable distribution of ability, effectiveness and fundraising ability of the nation's aid organizations raising the possibility of unsustainable aid programs given that resources are only available as a function of the generosity of the American public and is also therefore subject to the cycles of the economy (bad times, people less generous, etc) and 2) the mistargeting of problems.
2) is a bit complicated to let me explain. Aid organizations, especially religious ones, disproportionately target problems like food security, healthcare and shelter availability. These are fantastic things to try to improve however they, excluding healthcare, do not constitute the major expenses funded via taxation. Rather, roads, infrastructure, water supply integrity, electricity expansion and public education are also massive expenses that are important but less dire on the surface than illness and homelessness. You can go on and on about how many homeless people you fed but if you can't maintain the integrity of the water supply with regular maintenance then the majority of your population might as well be living in the slums of a third world country. Since these efforts would be entirely dependent on public charity it would simply be the case that these very important, expensive, and broad programs would go incredibly underfunded and the country would languish. It would not be sufficient to use "extra funds" from other efforts as the problem of healthcare and food security in this country are large and expensive enough to drain multiple times the coffers of whatever it is we could raise on public charity.
The other alternative is creating a national charity fund run by the government that people indiscriminately pour money into and then the main aid organizational body distributes those funds appropriately. However, then you have the same problem you have with government waste in current welfare systems. Namely, aid distribution is subject to adequate representation of a cause or community in the lrganizational body and this would lead to conflict and debate on how to best spend this money. You would essentially have precisely the same system we have currently funded with taxes just with less money overall. If anyone would like to make the argument that public charity would surpass the amount we currently collect on taxes then I kindly request they turn around before they make such a statement in order to ensure that they are speaking directly from an ass.
Then there's the issue of funding the American military enterprise which is responsible not only for American interests but countless foreign, private, public, corporate and charitable interests as well all over the world.
But what would be the upside? The American people would have more money to spend on "property" and "investment" and "capital" so we can have more goods and services that more people couldn't afford. Yay. Worth it, for sure, right?