More units or more affordable units would bring rent down overall
The sentiment is correct, but in practice (a) there isn’t enough land to build enough units and (b) the construction generally takes long enough that units not deemed “profitable enough” will simply not get built.
As for affordable units, some of the major drivers in cost for units is land acquisition and public service apportionment (producing water, power, and piping out shit and disposing of trash). As the density of communities increases, the cost to provide each increment of all of those increases. Additionally, providing new construction in dense city centers where other services existing (transportation, for ex) has higher than average costs due to the difficulty of working in proximity to other structures and active transport and to the increased cost of building vertically rather than horizontally. The alternative is to build lower-height accommodations in less dense/costly areas and have to recreate all of the services (transportation, shopping, etc.) in the new area, not to mention the inconvenience of being located remotely from the city center, increasing commute times for in-person workers.
Realistically, nobody is going to build low-cost housing in populous areas. There are enough people with enough money to saturate the market at high rates. Until that demand evaporates, supply will never outstrip it. If the owners could build more units cost-effectively they already would have. The owners of rent-stabilized real estate know the difficulty of building more units and just want to be able to charge “market rate” for their prime properties rather than go through the headache of building more housing.