Yes, I agree.
Perhaps a related point- in children whose parents try to pull them out of their local public school and enroll them in a charter school, but don't get in since they were not chosen in the lottery for the charter school-
Those kids have higher test scores than similar kids at the public school, even though the kid didn't actually enroll in the charter school and just stayed at their same public school.
Presumably parents who are organized/motivated enough to try to get their kids into what they perceive to be a better school do other things at home that help their child succeed.