You have bought the myth. The “public schools,” as now understand the term really dates from the 1870s, when Boston established the first district. Besides, public education doesn’t necessarily mean government run schooling. Virtually all the colleges and universities before the land-grant colleges were private and founded by religious bodies. Ditto the academies. Community schools were almost entirely elementary schools running through the eight grade. In 1900, only ten percent of the 17th year olds were in secondary level education. The majority were in academies that schooled their students through the first two years of college. St. Edward’s Academic in Austin turned out engineers as good as A&M’s. After 1910, the progressives began to push secondary education for all. Schools got very large and were operated pretty much like factories, using Taylor’s methods for efficiency. There was a push for vocation education, and until after the 2nd world war, the high schools did train students in usable skills. But after the GI Bill made a college education readily available, college prep gradually became the only real goal. Had to provide customer for the rapidly expanding colleges.