Popular government was considered risky (Hamilton believed it was bound to fail) and educating the masses was seen by Madison and Jefferson as the best remedy to losing liberty to the tyranny of the majority. They wanted a method of assuring that poor people would have the ability to become educated as well as a wealthy elite. Jefferson even came up with a plan to identify promising scholars from the masses and provide them funding for university education.
But Madison also understood and wrote in the federalist papers that time will be the ultimate test of policies and the great advantage of constututional and popular government is that it allows for correcting mistakes that will inevitably be made and which will be revealed as time passes. A policy that worked in the early 1900's may be rendered harmful in changed circumstances and there is no reason we have to follow the same state monopoly model of education just because that is what teachers unions want for their own selfish reasons.
The public school system today in no way educates citizens to be responsible participants in self-government. The harm it has done has been immense but it is not irreversible. Vouchers to stimulate the private education market, home schooling and removing government as much as possible from the lives of families and children is what is needed today to begin to reverse this harm. Madison would be the first to admit the need to back away from the one size fits all system of indoctrination that masquerades as public education today.