What would be the funding formula, and what would be a reasonable profit margin? Just like in other common areas of activity by the government (parks, garbage, policing, fire, sewers, utilities, emergency health care) there is obviously room for private enterprise involvement. The question always is who pays the companies providing the schools and how do you avoid jacking up the price of education.
Any large organization builds up bureaucracy and red tape (look at the top HMOs) so large education corporations would not be much more efficient than public boards (especially as time goes by). They would also have a care of duty to maximize profits for their shareholders while providing a competitive service.
What the health care mess has shown is that a mixed public/private system can easily incorporate the worst, rather than the best, of both systems. We would have to be very careful to avoid a similar mess if education reform went in a similar direction.