Alright, this topic is like old bathwater to me, as I and my wife were homeschooled, by parents who are radical countercultural religious homeschoolers, among a network of similar idealogues, and I even worked ten years for a well-known homeschool-textbook publisher. I think a couple of points are worth making.
First, there is too much mystery and misunderstanding around this topic. Somehow, some elements have taken homeschooling as a brand for their own movements, and the ruse has worked so well that homeschooling and the reaction to it has become defined by these elements. However, the truth is very different. For example, did you know that the largest demographic of homeschoolers at the moment is probably Roman Catholics? This has been true for a few years, as that church has been unable to maintain the level of education for which it had been known, closing down many parish schools.
Second, to return to the elements that prey (in my opinion) on homeschoolers for their own ends -- these are for the most part offering quite terrible curricula, academically, quite apart from the pernicious nature of their political and religious agenda. As they seem hell-bent on making the perception a reality -- that is, on in fact making all homeschooling merely the acceptance of their brand -- I expect the average quality of homeschool education soon markedly to degrade.
Third -- and here we do touch on a mystery -- yes, studies and experience show that homeschooled children perform remarkably well academically, but I think the reason is psychological and sociological, is even a reason of the soul, viz.: This is what children are capable of, really and truly themselves capable of. A parent is a small obstacle; an institution a complex large one. In other words, it is not, in my opinion, these parents or their curricula that are to be praised, but our modern institutional ignorance of the soul that is to be blamed for average American children's suppressed inner vitality.