I'm going to start this off in a very unusual direction for me - a Milton Friedman quote, which is now third-hand, from a newspaper editorial to me "Government excels at getting money and throwing it at a problem - government fails at spending that money efficiently"
Government money, allocated to private schools, would most likely be able to replace the public education system at similar cost, and would do so in a way that probably made schools, on the whole, better, and in a way that allowed poor performing schools to be ended and replaced with schools that could do better. The private sector is really, really good at eliminating what doesn't work, and so long as school performance was tied to school accountability, the system would self-correct and improve itself.
The problem with this is that even the fastest-moving industries take time to establish themselves and become efficient, and even given a perfect system at the end of ten years, any given school system will have failed a huge amount of kids. In a good-sized city, quite possibly in the tens of thousands.
Immediate self interest will work against any major changes in education, and while the long-term benefits everyone, the short term hurts too many people for it to be politically feasible.
Except, say, in a city like New Orleans, where the system itself was already terrible, and the opportunity exists for it to rebuild in stratling new directions. Charter schools ,not private or public mainstream, but still...
(Blog Post inspired by a segment from NOW that will be online sometime in the next week)