People will sleep when they have to sleep. And if they have a choice between outside or inside on a cold night, they will choose in. These people don’t disappear when you make their lives even harder. They have to adapt to further misery. This isn’t a solution. It isn’t even a solution to the problems you listed. Ride the train early morning or late night? You will see a bunch of homeless people trying warm or get some sleep.

Don’t like it? Spend money fixing it. Not making the lack of adequate current solutions an even bigger problem.