The conception of the public schools as a coercive and homogenizing moral force is fundamental to the mandatory-education project — our very first public-education law (known by the wonderfully evocative title “Old Deluder Satan Act”) was explicitly anti-Catholic in its intent, as were many of the public-education laws (Blaine amendments, etc.). Like our Puritan forebears, contemporary progressives believe that what keeps the infidels from the One True Faith is mostly ignorance, which can be cured through coercive indoctrination.
Leaving poor families with no choice in educational matters is, from that point of view, the great selling point of abolishing homeschooling and other alternatives — not a regrettable tradeoff.
We talk about the “separation of church and state,” but the Left is very much interested in evangelism and in using the apparatus of the state to that end. Homeschooling is one of the few authentically radical movements of any consequence in contemporary American life, and the desire of powerful people such as Elizabeth Bartholet to inhibit it is very strong.