Bingo!
And, let's not forget public schools are reflective of the socio-economic conditions of their districts. The wealthiest districts in major population centers like New York or Chicago have some of the highest rated schools in the country.
In fact, while the over-all numbers of a school system like Chicago's are mediocre to poor, they still have some of the best individual schools in the country. Which ones? The 'Magnate' schools, which are of course selective admittance!
When AZ private schools are forced to take all comers, including the slackers, miscreants, trouble-makers, truants, ill-prepared, ESL, culturally naive & unaware, socially disadvantaged, impoverished, nutritional-deprived, lacking in parental involvement or even lacking parents themselves, etc., etc., etc., I assure you the private school numbers will plummet!