That's not how it actually works in practice. Landlords are able to charge tenants as much rent as the tenant can afford, so tenant income increases are what causes rent increases.

Supply and demand don't "work" the same when there is a monopoly of a necessary good.

Consider the scenario where there is no more water in the world. All that is left is in a well in your back yard. You can charge whatever you want for the water, because people need it. The limit is what they can spend, because the demand is essentially infinite.

Now, if you had to purchase a more expensive pump to extract water from your well, would you charge more for your water? No, because you can already charge as much as you want.

I haven't seen the data for other countries, but in New Zealand, you can look at trends of rent vs house price and they are not correlated anywhere near the way rent and income is correlated.

Supply and demand do still exist and affect rent, but when supply is low for housing, landlord cost increases have no effect on rent.