I disagree with your contentions that education is a public good, that governments should provide public goods and that governments do it better - all three are incorrect.
Education is a private good best provided by a free market. All history indicates that is true as referenced by the book cited above.
Public goods should be provided by the public - first and primarily by free citizens freely acting together, secondarily as free citizens acting through their local governments, thirdly through the State and as a last resort - through the Fed. However, the authority of the fed is limited by the terms of the consitution - that is if we are going to have a govt that obeys laws. Self government means that the people should act themselves and govern themselves - and not always through the government.
Experience and history shows that markets provide high quality at low prices (always) and governments provide low quality and high prices (always.)