There are things in life we treat as core moral values even if they cause issues with incentives. For example, people should not go to prison for not paying their debts, even though it runs against incentives. Rent stabilization should be one of them as well. The rest of society should adjust.
Try to not pay your taxes and you end up in prison in all the countries I know about.
Anyway, I feel like your argument works against you, actually. I think a core moral value is to allow anyone to move into the city if they wish so. With rent control, you limit the privilege to live in a city mostly to those born there. This is worse than the unfair advantage you have when raised in a wealthy family. At least anyone can study and make a good life later on, but rent control doesn't allow for much movement altogether. If you're born outside the city - bad luck.
The same rent control alternative applies too. You can move to the outskirts of the city that are less desirable locations.
In fact, they did build housing. Luxury housing that doesn’t solve the problem at all. Student apartments that are so expensive to live in, you have to work a job while studying.
To build a ton of new housing, you need investors willing to do so. If you have a million dollars, would you invest to build in a rent control city, or in a city where you can raise the rent as you wish? Most people would choose option 2, thus creating shortages in rent control cities, which in turn make the housing more expensive than it would be if the rents would be free-market.