And it SHOULDN’T be state government, either. K-12 education should strictly be the purview of towns, townships, counties and cities. Centralized government, whether at State or Federal level, should not be in the picture at all.
And I disagree with you on what should be. Cities, counties and townships are creatures of state government. Education should be entirely private. Government (at any level) Education = Government Propaganda. I wouldn’t leave it to the Mayor Daleys either.