Actually, the major reason behind the public school movement in the first place was because of concern regarding religious teachings being provided in the private institutions that made up the school system of the time, beginning in 1818, when Boston became the first American city to establish a complete government-financed school system, and continuing up through the 1830s and 40s with Horace Mann and the rise of the government school system.
Before that time there were no tax-funded schools, so of course the founders would have no jurisdiction over those school boards.
When public shooling became popular the textbooks were McGuffey Readers and the Bible.
Before that time there were no tax-funded schools, so of course the founders would have no jurisdiction over those school boards.
Well, actually among the first instances of government funding of schools occurred with The Land Ordinance of 1785 in the prime of our Founders.