Some of those are rosy studies designed to be rosy tho. As in, many of those studies are are as independent as cigarette company paying for research on smoking and health. Also, a lot of effort goes into insulating religious homeschooling families from any independent sight at all.

The elephants in the room he mentions is very real thing too. Fair amount of homeschooling is explicitly social/political project. It is meant to shape both how family structure looks like (who is head of the house) and also meant to create young people that change larger society into religious one. ( Partly you can see it when you listen to homeschoolers talking about public school or even non religious people in general - frankly they often sound like aliens who got the idea from movies. )