Nah ... Schools are used because they have big facilities that are used only part of the day, lots of parking, and high visibility.
Churches are also used in both CT and IL. I have voted at Catholic Churches and Protestant Churches. I presently vote at the fire station, even when fire district funding is on the ballot.
I remember my elementary school and junior high schools were used as polling places when I was growing up. I have voted at a fire station, some other place I don’t recall, a few elementary schools in different states (not in the same election years! Haha), a casino on a reservation, and tomorrow will be a church building. One year, I registered to vote, but my registration was either lost in the mail or lost at the RAT run county elections office. It was found a week after the 2000 election. Hubby voted in that 2000 election at an elementary school.