I can’t speak for all of the US, but locally the most common way that affordable housing is built is because we have local government mandates which require it

I can’t speak for all of the US, but locally the most common way that affordable housing is built is because we have local government mandates which require it of large companies who build in my area. It seems to work very well; ours is the fastest-growing region in my state, and homelessness has been low.