I agree with you whole-heartedly on the goal and the reasons why this goal is desirable. The thing is, the way this is done currently, which requires ever-increasing rent prices, serves to drive away people who have their entirely life based on one city, destroying social links and causing mental health problems.
The solution is the decommodification of land, very high speed modes of renewable transpiration, and spreading around in more than two or three cities, as well as a massive increase in supply as public housing.
That way, we can achieve our goals. We can have big cities, that allow for low carbon transportation, we can keep lower income communities alive, we can avoid incredibly high amounts of wealth being wasted in non-productive housing, and we can have vibrant cities, for anyone, not just for the fortunate.
Yes, you could build more, and have 30% more people in your city, but with rent so high that half of the city lives in economic precarity or has to move out, destroying their entire social circle and access to the aforementioned benefits.
In the end, it we abandon the obsession on free market solutions and partially decomodify land, we can provide the big city lifestyle to everyone that wants it with the economic, environmental and cultural benefits it brings, without merely postponing the elephant in the room of unsustainable rent prices.
And by the way, according to the authors arguments, even if we tripled density we'd just roll back rent increases for a mere decade at best.
The government has two main advantages here. Firstly, it doesn't need to pay unsustainable market rates if it implements rent control before starting big public housing housing investments. This is a synergistic policy, because while at the same driving down real estate prices with rent controls, you counterbalance the issues of rent control which are decreased supply with your own increased supply. The end result is that now you have much more housing, for cheaper.
The other huge advantage is that while real estate investors expect a 6-7% YoY profit rate, the government is fine with 0% or even less if the increased economic activity can pay for it back in taxes. The real estate market is ultimately unproductive, so going from 6-7% of exponential growth to 0% in value extracted without productive returns is a huge economic gain in and of itself.
You do of course need to keep the government accountable to keep this housing of proper quality, but competition with homeownership or private landlords can help here.
The thing is, this actually works. Rent control tweaked to reduce supply issues combined with public housing to finish providing the necessary supply works to reduce rent and keep it under control while also allowing newcomers to enjoy life there too.
Market solutions on the other hand don't work on the long term. Eventually, rent just continues increasing. Look at Hong Kong for an example. The density went way up as the zoning rules relaxed , but so did rent too. Compare it to even the very bad Chinese Hukou system, where rent in Shenzhen or Guangdong or Beijing is much, much lower. You don't get to own land because it is decommodified, but you get to rent private or public housing at very low prices, and if you have the Hukou you also get to own guarantees on the land. Even the worst such systems in high density cities provide better results than free market solutions.