Rent controls absolutely limit supply, thereby making the ultimate problem worse. If you're a real estate developer, are you likely to spend money to build housing if you can't actually a charge a fair market price for it?
Rent controls lead to shittier quality housing. If you have an apartment that's priced 25% below market value, you're going to have people interested, even if it's in poor shape. So why bother with upkeep when it's not going to increase your return?
Rent controls lead to discrimination. If I have an apartment for rent, I'm probably going to let it go to the person who offers me the most money unless there are very clearly compelling reasons (e.g. the applicant is a convicted arsonist). If I can't use money to select a renter, I'm going to use other criteria, and that can get ugly fast. "Well, that young, single black man had good housing references and a stable income, but he might have 'thuggish' friends, so I think I'll go with that white couple".
Basically, it's a really, really bad idea. (I know you're not disagreeing, I just wanted to expand upon your points).