Where did the private sector fail at this?
It has not failed YET, but I firmly believe it would fail.
Private program management (at least in the transportation world) is good as an 'assisting' arm of the government, so the government does not have to permanently staff up, for certain projects. But when it's 100% controlled by private interests, that interest will take the place of the public good.
Certainly you have good government, bad government, good private companies, bad private companies. But you cannot deny that over time the private side WILL venture towards its own interest, while the government side MAY, and if so, only for a short time before the corruption is eliminated.