At least in Jefferson's eyes, if not those of Earl Warren, support for education was a federal matter, as well as a state and local issue (see NorthWest Ordinance, written by Jefferson, for mechanisms provided to finance education).
Going beyond the narrow issue of whether or not education is federal, state or local, in Jefferson's eyes, if not those of Earl Warren and that ilk, it was quite appropriate to rely on religious institutions, even, in fact, those held in low regard, even fear, for the purpose of providing education to the people.
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Was power over children's education delegated to the United States by the Constitution? Answer this question for yourself, and therein lies your answer.
for sure Thomas Jefferson proposed and Congress passed a budget that provided federal funds to Roman Catholic priests to teach school in the Old Northwest Territory!
That may not have been strictly Constitutional either (see also Louisiana Purchase). Your point?