While you are correct in that the Constitution does not directly address public education, and that the General Welfare clause is limited, Jefferson, Madison, Adams, Franklin, Webster, among others, believed that a basic education was instrumental to securing “life, liberty, and pursuit of happiness” and it enabled the American individual to “understand his duties” and “know his rights.” When the people are taught reading and history, they can then follow the news, and judge the best way to vote. An educated citizenry is best equipped to express themselves adequately to fight the tyranny of government. By providing equal access to primary schools, Jefferson hoped to teach children “to work out their own greatest happiness".
Jefferson supported the idea of public education, however, as you say, he would not have placed schools under direct government supervision. And while the government of Jefferson did provide support for the fledgling school system, he argued for the placement of “each school at once under the care of those most interested in its conduct.” He would put parents in charge:
"But if it is believed that these elementary schools will be better managed by…[any] general authority of the government, than by the parents within each ward, it is a belief against all experience.… No, my friend, the way to have good and safe government, is not to trust it all to one, but to divide it among the many, distributing to every one exactly the functions he is competent to."
Taxpayers were to provide the resources for public education (general welfare clause), and the community, and parents, would arrange the schooling. Unfortuantely the parents of today have acquiesced that authority, just as the people have given much of their authority to the government, and to fill the void left parties that have little interest in the good of the country as a whole have taken their place.
