Would it? I'm not so sure.
Wouldn't most if not almost all of the homeless just remain what they are and the new housing would be used by new people moving into the city? Unless there is a program actively moving homeless into the new housing, paying for them, and also a services to take care of mental health issues. For how many homeless would it work to just provide them housing, and from there they get back on their feet, into a job to pay for the rent by themselves, on their own? Compared to how many need a lot more active guidance, and how many will never stand on their own feet, for example because of health issues?
I'm pretty sure I saw this and actual real world experience and projects discussed here on HN, but I don't remember any details.
I think there is a lot more to it than just building housing, and that the topic deserves a closer look, for example here: https://www.feantsa.org/download/final_housing_toolkit_en-34... (PDF, Europe)
EDIT: Okay guys, please tell me why these thoughts are wrong.