The only way I see around this is a really aggressive cap on rent. Like, once a rent is established, it can never be raised, for any reason, ever again

We have something like that in my city. The rents grow, but are bound to the inflation. The dynamic is similar - the old contracts are vastly cheaper than the new contracts. That has several downsides:

• young families can’t afford good apartments, since they will need to tap the new-contract market which is expensive
• old people have nice large apartments for very little money. Sounds good at first, but then you often have situations where a single 80-year-old has a 100 m^2 apartment in the downtown, while a young 4 member family is in a 60 m^2 while paying more money. Even more perversely, the 80-year-old can’t move out, because they wouldn’t be able to afford a smaller sized apartment with a new contract.
• old contracts become a form of property in itself and are sold on “black market”, inherited etc.
• old contracts are not let go even if the apartment is not needed at the moment. So apartments are often empty, waiting for the kids to grow up, or used only occasionally (the family has a house outside the city and only occasionally spends time in the city).
• you’re heavily disincentivized to move (to be closer to your work, family, get a bigger apartment for growing family etc.) because your old contract would get cancelled, and you will need to get a much more expensive new one