Unless you want to build more sprawl-oriented social housing, you'd still need, in most American cities, to reform zoning codes and building codes (single stair and elevator reform) to get Vienna style social housing.
It's a "yes, and" problem though, mostly. Let the market build what it can, and if you want to pursue social housing, do that too - just don't let the perfect be the enemy of the good and delay the reforms while you try and put together all the social housing pieces.
No, you need a government agency which will build social housing.
Not once has the private sector ever been "encouraged" to build social housing with deregulation. The only reason it keeps getting touted as The Solution in the media is because deregulation would boost their profits.
The bottleneck is land, anyway. If you dont tax that enough or take it with eminent domain then you'll end up like San Francisco with absurdly low density housing and criminally high rents.