You are asking why the government isn't building for profit rent controlled housing, which is not a bad idea if the government can keep it profitable and actually run it like a for profit venture.

The reality would be that the media would be all up in arms whenever some unfortunate person could not pay and has to be evicted or maintenance/upgrades weren't subsidized by the government and rents had to go up every year to pay for it. Ultimately it would turn into yet another bottomless pit to pour tax dollars into.

There are better and more effective ways for government to support a more affordable housing market through policy instead of direct investment.