After a few years, they're certainly not getting paid the market price for their production.

Why not? When monopoly sets price it is market price, but when monopsony-like actor does same it is not?

versus just building affordable housing: the former doesn't increase the housing supply which means, even if rent is affordable, some people remain homeless.

I recommend you to watch Rossmann's walks around NYC where he just shows places that can be rented, but nobody does for 10 years. It is not because there is not enough supply, there was oversupply even before pandemic, just a lot of companies prefer to let place rot, then rent at ~~fair~~ literally market price because it will bring down rent on other places. Well, for housing there is also ban of everything that is not single-family shed or humant colony.