The real "problem" with public education in the States is the way most of them pay for it. School taxes are generally County taxes and taxes from the local school district; the states kick in a lot, and the Feds kick in a bit. As a result, school districts with a relatively low real estate tax base have less to spend, relatively speaking, than more prosperous school districts. As a result, prosperous school districts have more to spend on education, they pay their teachers more, they have more ancillary personnel, their facilities are larger and more robust. When a teaching vacancy comes up in a posh district, there are hundreds of applications; when they come up in a poor district, they go begging for teachers and hire less-qualified/desirable people to be teachers.
This is bullshit. The highest compensation should go to the teachers in districts where the students are the most problematic, not the rich districts where there are two college degrees per household, the kids have gone to expensive pre-schools, and parents hire tutors and private counselors when their kids have problems.
A "free" public education is guaranteed to the little tykes by the STATE, not the local school district. The State should provide essentially ALL of school funding, with a baseline objective of giving the same amount, on a per-student basis, to every school district, except where there are so few students that such a ratio would not generate enough money to operate a viable district.
The Feds have no legitimate role in K-12 education, except to the extent that students' "Constitutional" rights are jeopardized by the way the State fulfills its obligations("separate, but equal" and shit like that), in which case the DoJ might want to step in.
Nothing good has been gained by the Feds stepping in to finance "higher education," and they have no power to do so in any event. Anything they touch they **** up. Their involvement has resulted in massive inflation in tuitions for generations, to the extent that it is virtually impossible these days for anyone to work their way through college - an experience that is as fulfilling as the education itself.