But does it actually have that effect? Once everyone is using mortgages, everyone is back in the same place as before mortgages.

To put it more concretely: before mortgages, houses might average around $100K because the average person simply cannot afford any higher. Once mortgages arrive, houses might spike to $300K, because your average person can afford that mortgage. So while the example family might not have been able to afford the $100K house because they had $80K, a mortgage won't help because now houses are around $300K.

So I don't quite grasp how this would affect rents. That seems to be an orthogonal issue.