Quite the opposite. The benefits of rent control grow the longer you are in the same apartment without moving as the difference between what the tenant pays and the market value diverge further with each lease renewal. There are people in NY who have been in their apartments 50 years and pay 10% of the market rate.
Of course after multiple generations you scare off housing investment. But not after 5. And that should be the goal of rent control. Short term relief while doing the long term plan of building more housing.
Politicians not doing it this way is like blaming your duct tape for falling apart after a few weeks of adhesive duty.That doesn't mean duct tape is bad. It means no one bothered to fix the underlying issue.
Even when there's a plan in place, it's unpopular to remove handouts like that. Any politician up for re-election isn't going to let that expire.
You're assuming a form of rent control where new tenants pay market rate. That's not the only form, e.g., Berkeley's rent control used to continue "forever", until California forbade that (Costa Hawkins act in 1995).
They're right. Rent control is useful as a short term measure to keep rents from spiking, but it does long term damage to supplies and you need completely different methods to fix the supplies.