I was just thinking about this when typing up a top level reply. If you make things free, sometimes people will abuse it or do weird things - for example, homeless people taking sponge baths in public library bathrooms. But I doubt homeless people would clean themselves in public libraries at the same frequency if they had convenient access to "public showers" or some such.

But there are a lot of wrinkles to this kind of thinking. What if there is access to shelter, but it imposes some kind of deal breaking restriction like no drug usage? Or no felons? Let's get even more controversial - what if the shelter bans sex offenders, should homeless sex offenders be forced to shower in libraries? Things get complicated, and a lot of politicians aren't willing to die on these hills when their housed constituents have strong emotional reactions to these things.