This something only a government could really do.
Housing is nowhere near a free market. Public decisions control what can be built/rebuilt & where, essentially dictating "supply". It's also public infrastructure, services and wider economic policies that determine the value of that supply to consumers.
A quasi-laissez faire housing market can work. The US has many of them, in the geographically unrestricted cities like phoenix or.. lots of cities. The city/country builds roads. Land gets parceled into 1,000 - 5,000 sqm blocks and cheap houses are built on them, starting at $100k + taxes and land which are cheap relative to europe.
This can produce almost unlimited (given unlimited space) housing. Infrastructure is very basic: roads, sewage, etc. They usually don't have sidewalks, nevermind trams. They also produce low-density sprawl. As things get tighter, more and more burden is on cities to plan, provide infrastructure. Rights compete.
In theory, land efficiency and shorter transit distances should balance this out. In practice, land prices (enriched by infrastructure & proximity) generally outpace these efficiencies so house prices are higher anyway. Governments also limit supply in dense areas, because they can't keep up with the infrastructure requirements.
Truly revolutionary projects out of scope for private investors. First, horizons are too long. They can't wait 15 years for the value to emerge. Second, the governments' parts in the game (infrastructure & planning) is more important than private money's. Third, the money is too big, even for today's giant piles of cash. A small city (100k people) would cost 10bn to build, at least.
