Article I, Section 8, Clause 1 of the United States Constitution, is known as the Taxing and Spending Clause. It is the clause that gives the federal government of the United States its power of taxation and the power to spend for the general welfare.

Constitutional text
“ The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States; "

It's fully and unquestionably withing the power of the federal government to fund, and potentially control, education in this country.

I hate to tell you all this but your line of reasoning has for the most part won out (but obviously not enough to satisfy you) and has resulted in education in this country being controlled by an entrenched educational establishment whose policies are controlled primarily by teachers unions and other similar organizations. This has resulted in crappy, outmoded educational practices, that aren't much differant from what they were 100 years ago. If you want to do something constructive, which actually might come to pass, lobby the Obama administration to nominate Newt as Secretary of Education.