https://www.econlib.org/library/Enc/RentControl.html
https://www.aeaweb.org/articles?id=10.1257/mac.20170388
Beware of the anti-profit bias:
https://digest.bps.org.uk/2017/08/04/we-have-an-ingrained-an...
This would however go against a lot of interests, both institutional and citizens with a large mortgage, that depends on high housing prices. So it's rare. But one can see Vienna, Austria, as a good example of where the government takes an active role in building housing and can show a great result in low rents and high availability.
State-funded housing tends to be unimaginative/repetitive in design and lower quality though. The same outcome, of higher density housing, can be achieved by simply repealing zoning restrictions, and if that doesn't do enough, by replacing sales taxes with land taxes, which discourage sprawling land-inefficient housing.
There may be advantages, in political economy, that makes overcoming special interests opposed to densification easier, with the state-led approach though.
As for special interests in general, there are plenty in and close to the state itself that would massively profit from large public projects, so I don't see this as a question of whether or not we support special interests. It's simply a question of which special interests.
There are myriad reasons one area does great, and I don't think its entirely due to barriers in investing and building, else those with no barriers would see massive movement towards them. Often times the desirability of living in a place is almost directly inversely related to how little regulation they have.
Obviously density can compromise something like having good views, or the benefit of a relaxed low-density neighbourhood, but almost any other kind of housing scarcity can be addressed through densification.
The study in the aeaweb link above estimates that increased housing restrictions in just three cities in the US; New York, San Francisco and San Jose, caused the US to have 36% less economic growth between 1964 and 2009 than it otherwise would have, by inhibiting population growth in these cities, which all provide settings for higher productivity due to agglomeration effects and being coastal cities with low logistical costs.
A report by the White House's Council of Economic Advisors in 2016 largely came to the same conclusion:
https://obamawhitehouse.archives.gov/sites/whitehouse.gov/fi...
>>else those with no barriers would see massive movement towards them
We see a massive movement of people from New York and California to Texas:
https://www.northamerican.com/migration-map
It's hard to isolate factors behind this move, but I don't think it's a leap to assume lower housing costs, largely due to fewer restrictions on housing investment and construction, is one of them. In contrast to San Francisco, which is notorious for zoning restrictions and rent control mandates instituted by extreme-left city governments, and unsurprisingly, has the highest rental rates in the world (while not having the highest average income rates), Houston is renowned for its pro-free-market housing policy, and communersately affordable housing:
https://www.sightline.org/2017/09/21/yes-you-can-build-your-...
https://www.huffingtonpost.ca/2017/02/11/empty-homes-canada-...
https://247wallst.com/housing/2019/09/30/there-are-over-17-m...
But there is artificial scarcity in housing, and that's almost solely down to regulatory restrictions that suppress private property rights, whether that's rent control that make it less profitable than it should be to invest in housing, onerous permitting processes for building a new house, or zoning restrictions limiting the density of a new housing development. Economists have chronicled the problems created by these interventions, in detail.