It’s just not a problem capitalism is gonna solve. There isn’t huge profit to be made. Government could solve it.
In the US there is also reasonably cheap housing in large cities like Houston, Dallas, Atlanta, Denver, Chicago and Indianapolis.
In some large cities on the coast there isn't affordable housing. All these places have market economies. It's just that in places that are really successful if you have restrictive zoning and stop people building housing that people want prices go up.
It's also not just a US thing. In Australia, New Zealand and the UK zoning is also restrictive and housing is much harder to afford.
Allow people to build up and out and markets will provide affordable housing. Restrict the supply of housing and housing prices will shoot up in successful areas.
In New York and other cities, financiers and banks buy up entire apartment blocks and let them stand empty to drive up property prices.
The advantage of the government is it isn’t tied to a profit system.
You don't want to live in government housing when money is tight and some company offers a quick buck to pluck up that hole in the budget. Coops don't have that problem, they can't be sold, as they're owned by their members, which are largely also their renters.
A lot of places around the world have problems with affordable housing. In all those places attempts have been made to solve this issue. If this were an easy problem as many people seem to claim it is (I am not saying you said this) then it would have been solved already. It is not. The people running this fund realize that this problem is a lot harder than what it looks like on the surface.
The other problems with housing are solved pretty well by coops, even though I'm thinking of moving out of mine - providing affordable housing is nice on paper, but it also gets you neighbors that are socially challenged.
But the local government answers to local residents, but new developments don't help existing local residents (home owners), it helps new residents (often immigrants or young people who can't/won't vote).