What is the benefit over government of a single over-sized private bureaucracy. Aside from the tax issue, that would for all intents and purposes be even more centralized and inefficient. Private sector efficiency requires competition. I don't think that can be done to easily with charity.
It seems like it would do more to get people competative about giving. That would require strong consumer pressure to support charitable companies and avoid the noncharitable ones. If we prefer government incentive, we should up the rewards corporations receive for creating and contributing to useful charities.

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"Man lives in the sunlit world of that which he believes to be reality. But unseen by most is an underworld, a place that is just as real... but not as brightly lit... A DARK SIDE!"
-opening from Tales From the Darkside