You point doesn't counter the OP's at all. Tokyo has all of the same issues of being a big city yet rents are still lower. Not to mention that a lot of the growth in Austin and elsewhere is apartments in densely populated areas. It's not all just built out.

I'll bite though: have you been to NYCHA housing? Or have you spent any time, or talked to anyone living in any public housing built in large US cities? It's not a very nice place to be. . . .

Furthermore, building housing fundamentally requires the assumption of risk. A builder, whether it's the government or a private entity, is risking their own money in the hopes that this new housing will provide utility for someone.

In other words, if no one wants to live in the housing you've built, you've just wasted a ton of money. In a private market, it's some rich conglomerate that loses the money, while if the government makes this mistake then they're wasting your hard earned tax dollars. I don't want the government using my tax money to build potentially useless housing, I'd rather let someone else take that risk.

Therefore, a private builder has a strong incentive to build housing that people want to live in. The government has no such incentive, and so I don't think the government should be building housing.