Actually this is messy. There is not as strong a correlation between taxes and quality of schools as you might think although the high tax areas would have you believe otherwise.
https://www.usatoday.com/story/money/business/2015/01/15/247-wall-st-states-best-schools/21388041/
Number 5 and 7 on this list are zero income tax states. Also Washington (state) has great public schools and they are also a zero income tax state. Property taxes are also misleading because in many areas such as the northeast and Chicago, communities located along metra lines know there is demand for housing there and taxes are more inelastic.
In the case of Chicago specifically, they actually do have good schools, it's just very segregated. Evanston Township High School is a great school. And it's zoned to both high and low income areas. New Trier just to the north is one of the best high schools in America.
I think a lot of families opt to make this compromise: move to a low or no income tax state and choose to live in a high property tax zip code within that state vs living in a ultra high income tax state and within that state living in an even higher property tax zip code.