Well, here in the Heart Of Darkness we have what is in essence what you describe - but without the government middleman.
SZ has a very low tax rate, funding a decent level of public services - but not perhaps quite the range of social services and public programs - or boondoggles, depending on one's worldview - that other higher tax cantons do. It's a trade-off, one that the good Stimmburgers have voted for again and again.
However, there is a great deal of private giving going on. Those who can, and who wish to, do seem to be very generous in support many a worthy cause - both in canton and beyond.
Are all Congoese gazillionaires altruistic philanthropists? No, I'd guess that the percentage of hard-hearted barstewards is the same as anywhere else. But many, many people - gazillionaires and normal folk alike - do see it as an honor and duty to support the causes and projects they believe in. The low tax rate means that one has more disposable income to do so; Schwyzers seem to prefer the direct route when possible.
Do all programs get funded equally this way? No, not by a long shot. But should they? How should the funding-worthiness of a project be measured?
Many people would rather decide for themselves which groups and projects to support, and I would guess that when giving from the heart many folk offer support well beyond what they might have paid in higher taxes.
I'm most emphatically not of libertarian ilk. In fact, politically and philosophically I'm far closer to the pointy-headed liberal camp.

But I have come to appreciate the practical and effective Schwyzer approach.
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DB, a form of what you describe was proposed a few years ago; there was a feeling that it was 'unfair' that only members of the official churches paid the church tax, as the (debatable) idea was that that revenue went not only to building funds but also to (church sponsored) social programs. The proposal was that those who do not pay church tax should pay an additional 'caring' tax to help fund those social programs.
The idea was not well received - again, because I'd guess that people prefer to give as conscience dictates.