"The Department of Education supplies funding to States."
Nope, the money comes from congress, they're a middle man. They scrape a bunch off the top and then disperse it according to political pressure: Congressional funding could go directly to states, that can raise teacher salaries at will.
DoGE is cutting waste: money that doesn't contribute directly to learning. The Dept of Ed has not raised test scores since it was created. More power in Washington disempowers locals, including, of course, teachers.