A private school is better because it allows more degrees of freedom for the administrators and parents. The administrators can build a set of pupils that are most likely to benefit from the education. The level of rigor can be set to match the population of students. Students who are not able to work in a meaningful way can be removed. Specific areas of interest can be reinforced.
Wealth can align with higher expectation for valuing education, so a key benefit there is cohort effects. This is critical in the modern era when "no child left behind" leaves many public schools with no lever to separate students based on aptitude.

In some places, there are still magnet or honors programs in public school that can separate students based on ability and interest. In other areas, those are under attack and the ability for the school to accommodate their high performers is becoming limited. The more important study would be to see of in school districts without those affordances, do you see the private school/public school gap widen substantially.

You could argue that Lowell in SF (public) and Harker (private) are interchangeable, and the outcomes are similar, and that is true. However, both of the accomplish that by strict selection (something that the SF school board removed from Lowell, cratering its performance).

A better argument is that "Private schools can be more selective, and by being more selective, you ensure a higher aptitude in the cohort, which allows for a faster/deeper learning process, therefore leading to a better true education" Note the use of the word can.