Good post but it's not the "city" (as in the government)
It's nonprofit industrial complex, the NGOs trying to fix this problem get their money by the existence of the problem itself. if homelessness disappeared, where would they get the money? The city government can't solve the problem, so they use it for political capital.
Same reason the medical/pharmaceutical industry wants people sick enough to give them $$$ but healthy enough to keep them alive. You can't get money from someone who's healthy. Or dead. Only in-between.
It's all the same spin on the military industrial complex, an industry needs a problem to earn money for dealing with the problem. Perverse incentives for profit are incentives, that's all that matters. The colloquial term for people who see through this entire nonprofit scam is "poverty pimps"