Go check some of the major cities around America. They're graduating below 50% of the kids, and require over $10,000 per student. Not every school provides a great education. Heck, in my county alone, we have huge variance. The school my wife graduated from requires about $9,000 per student in taxpayer funding, and graduates about 81% of kids. Comparatively, the one I should have gone to graduates 93% of kids, and requires only $5,500 per student in taxpayer funding.
Let me throw something crazy at you:
What if private education is cheaper than public education? Where I live, it is. You know how much our local private school costs? About $4,000 per student - lower than any nearby school. That is why many that like private education believe in vouchers - that kids have the right to education, regardless of the facility. If private schools are really better, why can't you take the money you've paid into the system and send your kids to a better place if it costs the same amount of money?
I would expect if the quality of incoming kids is lower (less well raised) as is often the case in major urban areas in America, the cost per student would be much higher to make their test scores the same as a higher quality incoming student.
About homeschooling: I'd heard of studies saying that, on average, homeschooled kids perform better than public schooled kids. The problem with this statistic is that it doesn't account for widely varying qualities among public school students.
I dont doubt that most homeschooled kids have parents who care. This parents are often well off since theyll need the time to teach their child. So these homeschooled kids with devoted parents are being compared to public schools where many of the kids have extremely overworked/poor/not caring parents.
I wouldn't doubt that scores would be relatively the same for children from similar families regardless of where they go to school.