I fear that homelessness is conveniently, for some, too complicated (like climate change or gun violence) to address beyond the most basic of impacts. If a single solution that can cost-effectively solve 100% of problem can't be found, it will circle the drain before dying.
Heck, 1/2 our country seems to think that a soon-to-be convicted with a nearly 100(!) indictments hanging around his neck con man should be pulling the levers of the most powerful country in history.
However - that doesn't mean we ever stop trying to create a little balance and your idea of a progressive tax is a good first step.