(This is not contrary to you, I think):
See https://www.fresheconomicthinking.com/p/the-auckland-myth-th...
Let's start by defining "cheaper" as "cheaper rent" specifically. Rent will only decrease or stabilise when supply outstrips the absorption rate of the market. In New Zealand, for example, at least one analyst sees the equilibrium absorption rate of rentals to be 2.5% vacancies to total rentals. This means that to make rents go down (or stop going up at the same rate as household income), the available rentals must be greater than 2.5% of the total rentals.
One way to achieve that is to build a whole lot of housing and rent it out, far in excess of population growth in a city. To enable building a whole lot of housing, you can upzone a city. However, increasing potential housing does not actually increase the amount of housing.
To do that, you need to actually build houses. Now, anyone who wants to build a house will usually only do so if it makes financial sense to do so; essentially, when the risk-adjusted return on investment is high enough. Upzoning doesn't really change the risk-adjusted return on investment for developing a house.
Now, lowering rents WILL change the risk-adjusted RoI for building a house; it will LOWER the RoI! So builders will stop building when rents start decreasing, because the financials stop making sense. This is a natural hand-brake on (private) house-building rates, regardless of upzoning. Build enough housing to lower rents, and house builders will stop building houses!
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The kind of counter-factual we are looking for is not really expensive cities that become cheap as density increases, because density brings about more jobs, income, competition, and so on that mixes in with housing data as confounding variables.
Instead, we want: cities where rent as a proportion of disposable income goes down, while increasing population and density. Most likely, any city in that situation is going to be increasing housing units faster than population growth.
The next part of the analysis would be to figure out how they managed to do that despite absorption rates being a handbrake on the market :)
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A side-note; Auckland, which did upzone in ~2016 and again more recently, has not had a notable reduction in rent as a proportion of household income, despite increased density. Population growth meant that it hasn't really beaten the absorption rate of the market, except during Covid (lots of house-building going on at the same time as population decrease).