I think one reason they use churches instead of schools is because churches are by and large vacant on Tuesdays, whereas at schools you have to deal with having random voters in the same space as minors. There's safety and practical issues.
I vote in the basement of a church, and it doesn't even feel like a church. There's nothing there to indicate religion -- just a kitchen, an open space and a food pantry on the wall. I don't think the church gets any money for it. It's just a public service they provide. There just wouldn't be enough civil and public spaces if churches stopped allowing it or were banned from doing it.
I suspect justajane has given the most common scenario. There might be a handful of counterexamples among the thousands of polling places, but it's unlikely that voting occurs in "sanctified" worship spaces.
If we are to move to a more inclusive society, people may need to deal with their own versions of "I'm uncomfortable [with] __________ --find them slightly creepy to be honest-- and don't feel like I should have to [have anything to do with them] to do my civic duty."
It's easy to agree with that statement when it matches one's own "uncomfort zone," but not so easy when one disagrees. E.g., substitute "religious people" or "gay people" in the blank. In either case, I respectfully suggest the uncomfortable ones build a bridge and get over it.
I think it's easy, when you're not the one uncomfortable, to tell others they need to get over it. "Oh, you're uncomfortable? Well, I'm not. So too bad, you need to get over it." I personally think I've gotten over it as much as I need to (I go to said place, despite my discomfort, to do my civic duty, and I don't say anything to the good citizens who are helping at the poll booth, and I've thought through the practical implications of what might be, in my opinion, a better solution and have decided it's probably not).
And I didn't say "religious people," I said "houses of worship," so it's not the equivalent of "gay people" (also, have never had a gay person try to convert me, but whatever). I stand by my right to feel uncomfortable in a building that is laden with pictures and statues of a dead guy with blood running down him and who many people have insisted on telling me
died for my sins (oh no, he didn't!). And I believe I am entitled to feel uncomfortable doing my civic duty in a building/institution that has, repeatedly and with heavy-handed tactics (in my opinion) overstepped the line between separation of church and state.
I would be interested in see what would happen if more people were asked to vote in mosques, or houses of devil worship, or Wiccan sacred space. I believe I have built the bridge I need to and I do not act on my uncomfortable feelings--I do my civic duty--but my feelings are my own to have if I want to.