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.
Blaming "fake socialists" or "nimbyism" doesnt change the fact that they are entirely rational actors in an economic system that is rigged to make the wealthy wealthier (e.g. with prop 13).
In fact, it's a clever rhetorical device to conceal the root cause and displace criticism towards the main culprits and, well, you fell for it.
Some of them are even renters.
No one is trying to conceal the problems with unfair tax systems - indeed, the people who spend time advocating for housing are probably your most likely allies in trying to fix it.
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.