Counter-proposal:
Replace the housing voucher with a moving voucher.
If you can’t afford to live somewhere, find somewhere you can. Big cities are expensive for a _reason_. The biggest complaint I get about “move” is “they can’t afford to” - well, they can’t afford to live where they are either, so instead of trying to keep them there (which increases prices by increasing the supply of money available to pay for it), give ‘em a one-way ticket to wherever they like, moving van service included.
I'm not so sure about that. Trump holds the White House because a ton of young lefties are concentrated in small geographical locations. I kinda don't want to see them diluting all those safe Republican districts.