That assumes rent controlled units actually benefit mostly the underprivileged. They don't.

They create a strong incentive to keep a flat indefinitely. It's not unheard of for a person to continue renting a rent frozen apartment after they buy a house because the rent is so cheap - effectively it's a second home.

There's also plenty of corruption. "Knowing a guy" is one of the best ways to get a rent controlled unit since the wait time can be years.