The dirty secret of education is that no school outperform nor underperform much other schools when adjusting for differences in student populations :

https://freddiedeboer.substack.com/p/why-selection-bias-is-t...

There’s simply no private schools in poor districts to serve poor families. I am not sure how do you run the comparison.

Boarding schools exist, as do affluent areas next to poor ones.

Anyway, there are several ways to compare systems, some private schools operate in whole or in part on a lottery system so you can track students who do or don’t get in. But the most common method is to model parent education and income as a predictor of performance and then compare outcomes.

As you suggest poor students are underrepresented in private schools, but some do get in. The important thing to remember is a student who received a scholarship isn’t representative of the general population.