Am I missing the obvious here, a smart and nimble company can corner this market by lowering their price by X. They would have to give up X per unit but they make it up by (volume * lower_price). Isn't that how the market is supposed to work? Like if I want all the business and there's not an obvious differentiator I will compete on price?
That's not how it works because there is no volume. You can't provide more supply if there's no supply to provide. Each building has a fixed number of units and the management for those buildings are trying to maximize the value they get out of each unit. Most cities have a mostly inelastic number of units and population grows every year, so you can pretty much charge whatever you want and people will pay it since there's no alternatives.
Why doesn't someone just build a new building? Mostly they will be blocked by NIMBYism. NIMBYism is a near universal philsophy, to the point that renters who would benefit from this are typically NIMBYs too. Sometimes they're the loudest NIMBYs, It will be pretty much impossible to build anything new until the state takes control of building away from localities and forces them to allow building.
This assumes vacancy is at 0%, for cities like SF I can see this, but do all cities have near 0% vacancy rate? Genuine question, I don't know. I would offer at lower price and have all my units filled.
Typically the price fixing part is one small feature of a large software suite, and deviations require written justification submitted to the provider (with the implied threat of losing access to the management software).