How, exactly, is inserting a middleman with probable ulterior motives going to work better?

Granted, there is no perfect system, and there would almost certainly be abuse (in any system). However, I think that organizations that are already doing the work because of religious or humanitarian convictions will do a far superior job at providing better care in a more efficient manner because they were motivated to do so whether they have the funding or not.

I also don't view it as adding a middle man. Giving to external organizational structures negates the need for the government to duplicate those organizational structures internally. The internal government structures tend to be on the inefficient side.

To take another area of government as an example, I work with educational administration software for a living. According to the National Center for Educational Statistics, the average expenditure for public school students is around $8,000/yr. For private schools, it's a little more than half that. Yet, private school students consistently outperform their public school peers in standardized testing. That and dozens of other areas where I've seen government inefficiency in effect lead to my views on goverment involvement.