The problem with these types of price controls is that it’s inherently unfair. If you’re on the inside and lucky enough to get a rent controlled apartment, you win. Everyone else loses.
Or said another way, if you have strong ties to the city and/or are a longtime resident of the neighborhood you win, if you don't you lose.
This has literally nothing to do with who ends up in a rent stabilized apartment.
In fact, new residents are more likely to end up in rent stabilized apartments than existing residents, so rent stabilization has the exact opposite effect of what you're describing it does.
Citation needed on that one.