It didn’t control for wealth though just income. It also didn’t control for parental education or familiarity with elite colleges. It didn’t control for parental expectations (seems likely that a parent who spends money on private school will be more likely to expect their child to attend an elite college).

It didn’t control for the value parents put on education in general.

It didn’t control for children’s expectations (seems likely that a high school student who wants to attend an elite college will be more likely to push to attend private school).

It didn’t control for extracurricular activities (it seems likely that private school parents will be more likely to encourage their kids to participate in extracurriculars that look good on an elite college application).

It did control for SAT scores through, so the one thing this study is definitely not doing is examining the effectiveness of public vs private schools at effectively educating children.