Not sure why you'd think OP is saying 1 church per town. Just that there are ~380k churches in the United States, and less than twice as many homeless people.
I agree, far too many people are left out in the cold at night when we have many public, climate-controlled buildings with working bathrooms and possibly even showers that are empty after a certain hour. If the homeless were able to regularly get a good night's sleep and a shower in, they might be more able to hold down jobs and become contributing members of society again.
Schools certainly would be great as a shelter after hours, most have gyms with showers, possibly laundry machines, and certainly ample space for someone to sleep with a sleeping bag. If we could just figure out a way to make sure everything stays clean for students to use the next day, no left-behind drugs, no vandalism, etc. that could be a wonderful solution.
My guess is that in most places the homeless population would easily fit within the gymnasium alone.