Only in New England could this claim be made where there were public schools. That is what Adams may have been referring to since he certainly did not believe that universal literacy existed in the south.
As late as 1860 there were large areas of the South where illiteracy was predominate. Of course, the author of this piece of screwball crap somehow "forgets" that it was illegal to teach slaves how to read and write. Thus, the majority in S. Carolina were forbidden by law to learn to read as well as sizeable minorities (30-40% of the total population) in the other states of the Slaveocracy. Travelers to the region during the 1800s repeatedly commented on the lack of culture, even the lack of books. Southern aristocrats looked down on educated men and felt little need to learn anything not related to raising dawgs and hunting,hunting, growing slaves or crops. They would not allow their sons to learn how to play music or paint. Novelists from the region would publish under an assumed name to escape the disrepute literary achievements entailed. It was precisely the lack of educational opportunity which drove many of the leaders (Madison, Calhoun etc.) to northern colleges.
About two-thirds of those at the Constitutional convention were college graduates and lawyers to boot. Our nation's early leaders were frequently college graduates. Only Washington did not have a college degree of our first six presidents. Jackson didn't but that is likely due to his being an orphan. Until Lincoln almost all non-college graduate presidents were military men. I suspect that most of the Senators and Representatives were college educated as well.
What passed for education for the vast majority of people was an ability to read a little. What passed for learning (other than classical languages and Bible studies) was pretty much laughable and wouldn't get you very far today. As far as true learning or culture went there was very little of it in colonial america or after the revolution until the 1800s.
Since generally only the sons of the wealthy were able to have private tutors (another lie that there was a lot of home schooling it was mainly through private tutors) the movement for public schools led to their establishment in the North but the South wallowed in ignorance until the War made it clear that the lack of education was a major factor in its defeat.
Why does such mythology gain acceptance around here? Our founders do not need ridiculous claims to raise their reputations. It does not discredit them to accurately describe them as highly educated men as most of the leaders were. Particularly when compared to their fellows.