If this was some super-costly policy that needed a big apparatus around it, then they'd have a point, but e.g. in Finland, one estimate is that it costs them up to 9,600 euro a year less to house a person first vs. leaving them homeless. As such, just starting to provide some housing units and gradually grow it would be a win for every local government with a homeless person.
It only starts to become a challenge if a few local governments reaches such a level of provision that it attracts homeless people from surrounding areas that don't do anything themselves, but that's not a reason not to start.
Sometimes it feels like US taxpayers wants the government to burn money if the alternative is to do something that might help other people with it.