There are basically two arguments here.
First, you can’t just open the door to a heated building and say it’s open for all incoming ‘guests’. There has to be organization to this, certain limits on how many can be safely housed for the evening, and sanitation efforts detailed out. For some church operations, they haven’t done their homework, or understood what they are volunteering to do.
Second, city governments usually do a crappy job when they stand up and suggest they will run such-and-such program. In the cases where churches have been involved and gotten support from it’s membership (qualified managers, technicians, craftsmen, etc)...they’ve normally done a better job.
I belong to a church that is housed in a hundred year old feed store. We had a homeless old carpenter that did work around the place and slept in his truck. Gradually he built himself a nest in the wet, cold dirty basement. He stayed on and off for a couple of years. Technically, the building was not zoned or equipped to be a domicile. This is a safety issue, not a church/state issue. It would be better for the members to either have these people in their homes or sponsor them to stay in an adequate facility. There could be a NIMBY component to this, but I believe the city is right.