I am of the opinion that at the federal level, there are only two constitutional education "authorities" for the government.
1) FOR THE PURPOSE OF HIRING GOVERNMENT EMPLOYEES, determine that constitutes an approved GED, HS diploma, Associates, Bachelors, Masters, Doctors or equivalent. This could be handled by a small office out of the OPM
2) FOR THE PURPOSE OF TRANSFERRING AN EDUCATION LEVEL BETWEEN STATES, determine what constitutes grade equivalence. I.e. Fifth grade math means ...., Seventh grade science means .... HS diploma means .... Again, this could be done by a small office. Under this authority, the federal government could publish a grade specific, freely available via the Internet, course curriculum for each grade. Schools, both public and private as well as home schoolers, could then down load the curriculum and use IF THEY CHOOSE. All would still be free to use whatever curriculum they chose.
No federal student loans or grants
No federal regulations regarding sports
No other federal authority
And most certainly, no need for a cabinet level position.