Very interesting info. Thanks, Cutlass.
Funnily enough, we have rather the reverse problem here. Specifically, a lot of private landlords rent out to council tenants and the council pays (most of) the rent as housing benefit. At the same time, there are national level policies that inflate the housing market. This leads to high rents, based on high house prices, while tenants have less reason to negotiate since someone else is paying; and as many of the people in receipt of housing benefit are in work, it also puts downward pressure on wages, since these need not cover the full cost of rent.
So it's very damaging, unless you're a landlord or capitalist of course, but to untangle this mess, unless done very carefully, would cause serious hardship to many poorer people. It's a lesson on the dangers of mixing public provision with private gain.
In the US shortages of appropriate housing are almost always caused by local government regulation and zoning laws. They exclude multifamily, apartments, and small houses from areas to keep the price up. And that keeps the poor out. There are federal, state, and local housing assistance programs. But it can take years being on a waiting list to get into a program.