” Congress would probably enact it as a statute”
Such a Supreme Court would even more quickly toss THAT statute out, because the Constitution does not give the federal government any regulatory power over education — unless the educational institution receives federal funds.
“Can’t he (or she) do this via executive order?”
No. See above.
Multiple Supreme Court cases give parents the right to give their children religious education. States (not the federal government) CAN set standards and enforce them, though, although not for the religious part of education. New York has been regulating private educational institutions since the 1780s.