Rent control doesn't work because of liquidity preference. Rental income must exceed the liquidity premium of land or else it will be kept empty. Liquidity preference is effectively the concept that people prefer keeping assets exchangeable for other assets.

For real estate this means keeping the property ready to sell at any time. It is easier to exchange an empty apartment and even easier to exchange an empty plot than a rented out apartment.

The downsides of renting out, i.e. the inability of being able to quickly sell the property must be compensated and that compensation is part of your rent. If you cap rent, that compensation will be insufficient to "bait" the property owner into renting his property out. They would rather keep it liquid and wait for a better offer. To keep it simple, the owner has options, the renter doesn't, so the renter must go out of his way to attract the owner's attention by paying higher rent than if he had outright owned the property himself.

Eliminating the liquidity preference of land would require all land to be leased or a sufficiently high tax on owned land (so that waiting and doing nothing has opportunity costs).