Public education began its long decline long before the federal government got deeply involved. And conservative southern states were often educational laggards decades ago (although one difference is that these states were run by conservative Democrats, now, there are more Republicans in power).
"Public schools used to do a good job educating children."
In some places, they did, in other place, not really.
sitetest
While many parents in the 1800s and early 1900s could have taught their children to read and write, I doubt they could have done as good a job as public schools teaching beyond basic skills down South or elsewhere.