If inflation is 10% a year for 100+ years you are going to have bigger problems than the rent 130 year old person is paying.

I chose that number for easy math. The point is that rent grows less than half as quick as inflation, which is definitely unsustainable.

That is if you don't renovate nor modernise the building at all. It is an incentive for landlords to keep the building up to date. In your corner case scenario, an outdated 100 years old apartment would be pretty lame and worth its fair price of "almost free".

Essentially yes, that's correct.