The problem is some places are more desirable than others. Penthouses, even if they're not big, they are, for most people, more desirable than the bottom floor. They have a view. They're away from noise. They're often away from rats, roaches, ants which start at the bottom.

Similarly any place on the side of a hill, a cliff, a beach, ... You're unlikely to ban such places (all buildings 1 story?) Even if you managed that they're the house on the quiet street and the loud house next to the highway/trainline/airport

You could try to make such places merit based but that only turns into a different kind of bribe and cronyism. Money is at least, somewhat obtainable by anyone. Plenty of people were not rich and became rich, maybe by luck, but at least it's a higher rate of churn than central authority has ever been.