I always find it a little disingenuous when lists of "famous homeschooled people" are passed around. I guess you can say that Thomas Jefferson was homeschooled, but there wasn't much in the way of public school in the South 200 years ago, and if his family was wealthy it would have been the norm for them to hire a team of private tutors for him and his siblings. The boys would have been prepared to go to college or some sort of apprenticeship (in the mid-teen years was the norm, and the apprenticeship would not necessarily have been manual, but something like law), and the girls would have been taught enough reading, writing, and math to be able to run a household. If you want to call that home schooling, I guess you'd be technically correct, but it seems kind of desperate to me to need to claim that as some sort of validation for what you're doing now.
I see that sort of claim on lists and t shirts all the time, so obviously people identify with it, but it kind of completely undermines the modern realities of homeschooling and why people choose to do it, and the cultural norms that existed for privileged sons on vast estates in a time period pretty foreign to our own.
There was a thread on this very recently, in the past week or so, but basically my answer is that public schooling began in New England in the mid 17th century because it was very important to the Puritan faith that individuals be able to read the Bible themselves. Then the US was founded on the Enlightenment ideal of a rational, educated populace participating in a democracy and public schools were and continue to be the only way that ideal has a fighting chance. For practical reasons, the public education system evolved in various different ways around the country, but firmly took root during Westward Expansion when laws were passed ensuring public schools for settlements.
As for how many mothers on the prairie had degrees in literature in math, the answer would be not many. But are you insinuating that poor, rural 19th century children were particularly educated by said mothers? Because I'm not sure I'd agree. Such kids would be lucky if they could read and add single digit numbers. If you're referring to that "graduation exam" that goes around every once and a while that supposedly shows how educated 6th graders in Kansas were a hundred years ago, I believe that Snopes covers it.
I dunno, I guess I would concentrate more on your own reasons for homeschooling, instead of trying to make broad generalizations about US history that don't really mesh with the current reality. If you're looking to be able to quote modern facts and figures about the shortcomings of the modern public school system, my old standby is The Schools We Need, by E.D. Hirsch.
Trying to live a simple life in a messy house in a complicated world with : DH, DD (b. 07/07), DS (b. 02/09), and DD (b. 10/10)