Because it is in a large part the means in which Americans are controlled? As we've seen with mass media, when control of everything falls into few hands, the special interests take over those people in no time flat.
The DoE passes down mandates and standardized testing, as well as other control programs, down to the states. Politicians and bureaucrats are deciding what your kids are taught in school. Ones who love big government, welfare state, policing the world, ect.
I think our society today is a perfect example of why you don't want the government running education. Firstly people end up stupid and uneducated, since central planning never works and educations no exception. Then you also get political and ideological views being passed to kids at a very young age, which they carry on with them as dead cold fact as the rest of their lives. Like how government is there to help you and solve your problems, instead of the truth that government is almost always the cause of your problems.
So yeah, we need parents and local communities deciding what they teach their kids, not DC 'good old boys', ie rich globalist elitist businessmen and the politicians they own. He doesn't even want to abolish public education like more extremist libertarian factions do, he just doesn't want all the money and power and decisions coming down from our benevolent overlords in DC.
I remember when I was in elementary school and we talked about the first gulf war and the somalia interventions in class. We had articles from newsweek and newspapers and such to go on, and a teacher telling us about it. Not even once was the legitimacy of the war itself or whether or not America should be getting involved in the first place ever talked about. The concept of intervention? Never brought up, it's just assumed that if someone thinks someone else is doing something wrong or bad, America rushes in to the rescue. I bet kids for years have been being taught the exact same thing about Iraq and Afghanistan.
This is why so much of america has no idea what the ideals of the constitution are. You are taught what the constitution is, and the details of it's drafting, but nothing of the spirit of it. Liberty, real money, non-interventionism, small restrained government, none of that is ever discussed anywhere in public (or the private school i attended for a few years) schools these days, and for a good long time. After the DoE conincidentally
Hence an entire country filled with people who think they are free while a small group of people run everything, borrow endlessly in their name to finance their re-elections and dreams of empire, meanwhile debasing the currency and regulating and taxing the jobs out of the country.