Public schools have existed in this country since before there was a US. Even Thomas Jefferson was a supporter of publicly-funded schools.
And publicly funded universities have existed in the West since at least the 11th century, with the founding of Cambridge and Oxford.
You should read Benjamin Franklin's treatise on public education. There is nothing even close to that today. The atheistic Secular Humanists would be in court in a nanosecond if anyone tried to get government funding for it. I expect Jefferson felt similarly about education. .
And,.. yes,.. you are correct that there were scattered communities that did compel the education of children but homeschooling was also completely acceptable and the standard for education was basic literacy and very basic arithmetic. Two to three years of education ( sometimes only a 2 or 3 months out of the year or attendance at Sunday school after church once a week ) was sufficient to fulfill the law. Most often those community schools were church owned and staffed.
The modern system of compulsory (police enforced) government owned and run schools, as we know them today, did not get a solid foothold in the U.S. until the mid 1800s. It was not established in every state of the Union until the first decade of the 20th century.
Harvard, as you might know, was once a church school, as was many of our earliest universities and colleges in the U.S.
As for Cambridge and Oxford...I bet the publicly funded Catholic Church and later the public funded Church of England had absolutely nothing to do with it. ( yeah right!) (sarc with eyeroll)
I suppose that you can be forgiven for your ignorance, but both of those institutions were founded by the Church of England, which happened to also own the government at the time.