The strongest argument for compulsory schooling probably came from Horace Mann , who gets a lot of the credit for creating public schools: for a democratic republic to function, the people need to be educated in order to make intelligent choices. And, the education should be at state expense, because that was the only practical way to keep it from being controlled by churches ( the older mode of education) and a sectarian education divided people. This combination of common sense and patriotism overcame most organized opposition. However, when he was making this argument circa 1840 most of the population of the US was rural and a great percentage of that were farmers. Farm children were often little more than domestic appliances, cheap labor. And even in trades- blacksmiths, housewrights- child labor was quite common. The use of children for labor was for a long time the real source of opposition. I have often heard ( would like to know a source) that the only reason there are three months of summer vacation for schools is a compromise from the days when farmers would need all the help they could get in their busiest months, i.e. give us your sons to educate for nine months, we'll let you use them on the farm when you really need them. Given the amount of child labor in the US in the 19th c., it seems at least plausible.

I don't know much about the debate over how high that education should go.