Suppose this is true. Then you replace that portion of the housing stock with new construction, and what problem is there? The problem is only if you're constrained from doing that.
> The other thing is that rents going up dramatically is a global phenomenon
There have been two major recent drivers of housing demand. One, the interest rates that until COVID were near-zero (which is a global phenomenon because capital markets are global). Two, since COVID, work from home causing people to move around and shift where the demand is. Also a global phenomenon.
Both of these can drive short-term increases in rents on the order of a few years, but they should also drive new construction, and then once the construction has caught up to the demand the rents would go back down.
Unless the construction isn't allowed to happen.
> The argument that high rents are mainly due to large regulations and lack of building is largely not supported by the evidence.
It very much is. You can take any two cities and account for things like median income and population density and when you're done you'll consistently find that the one with more restrictive zoning has higher rent.