In Canada the federal government is giving lots of incentives to build more housing. Basically stuff like rEmOvInG ReD tApE so the private sector can build more.
It'd be great if they actually spent money building housing. But it's better than nothing, I guess.
The thing is that red tape also applies to public housing, too. Those incentives are for municipalities to remove local land use policies such as SFH zoning, parking minimums (like Toronto, Montreal, and Vancouver all removed recently), policies which would apply to public housing projects, too. Removing these NIMBY land use policies is a necessary pre-condition for both public and private housing.
In my old neighborhood, higher density new builds were more expensive than the single family homes they replaced. Historically, housing was most affordable when the provincial governments (with the help of the feds) built housing. I'd like to see that start again.
Increasing density is great for a bunch of reasons we probably agree on, but I don't see it improving affordability without external pressure.