My problem with your argument is that you're treating them as three monolithic options. PUBLIC, PRIVATE, HOMESCHOOL. Well, there are some awesome public schools, and some terrible ones. Some are great for some kids but not others. There are some really bad homeschoolers and some really awesome ones, as well as some kids who wouldn't thrive in homeschool even with the best most well-prepared parents ever. Sometimes a kid would thrive in homeschool but the parent isn't suited to it. And private school- well, there are good ones and bad ones, just like public. Even if it's a fantastic private school, sometimes the expense puts a burden on the family that negates the benefit offered by the school.
So, to me, you really can't say "Homeschooling is bad" OR "Public schools are failing" and be right about any given homeschooling family or public school. Especially now that there is more choice given to families about where kids can go- we chose a charter school which is SO much more in line with my philosophy of education than any of our district's public options, and yet it's still a free public school. When my daughter starts middle school, my son will be entering Kindergarten we'll have two more decisions to make. If we can't make the choices available to us fit our life for both kids, it's very possible that I would homeschool one of them and send the other to an outside school (probably a charter).