I think there's a lot of secondary effects tied up in your statement that would not necessarily unravel the way you would wish them to. Ad absurdum, I don't see how it results in anything other than assigned, identical housing.

Because, even at a fundamental level, someone will always be willing to pay for something more than the bare minimum. And as long as that is true, there will be stratification. And that stratification will price individuals out of an area.