Earlier quoted context omitted.
The innovation here isn't the automated pricing, but convincing all the landlords in an area to collude by sharing pricing and demand data with a central organizer. Humans could do the pricing and it would still be illegal collusion.
If they are really facilitating collusion, then they would be selling a broken product. For example, for simplicity's sake, let's say that a city has a lot of landlords, but all their units are identical and they all follow the RealPage algorithm for price suggestions. Realpage suggests that they charge an exorbitant $10000 per unit, which leads to only 50% of all units being occupied, so your expected value of $5000…
You could instead imagine a (still completely fictitious) figure that makes your argument look bad -- e.g., that the raise in rates causes a 0% drop in occupancy. Then the landlords/algorithm win by raising prices.
One reason occupancy rates might remain relatively stable even in the face of rising prices is if consumers have relatively few better options, because e.g. all of your competitors collude to also raise their rates similarly.