Maybe if housing/land didn't appreciate over time then investment dollars would stop chasing it, and the overall price of renting would stay flat or decline due to the lack of capital inflow to the underlying asset. Who can say?
The problem with these thought experiments is you have to assume some reason why housing prices have stopped increasing, but also that nothing else about the economy has changed. Otherwise the thing that stopped housing prices increasing has probably also had some other pretty serious effects on the economy.
It's like asking "what if World War 2 never happened"? Well, why didn't it happen? The same forces would have pushed us in the same directions. If it didn't happen, well, that world doesn't look very much like the one we live in, so who can really say what would have happened?
In general, making as tiny a change as possible, housing still would be priced the same as it is now - i.e. based on what the market will bear, rather than some weird idea that landlords would get vindictive and crank rents just because their asset is no longer appreciating. To the extent possible that's what they already do, and they would price themselves out of the market with further increases.
But if landlording is still a revenue-positive activity, then people will do it, and housing prices would continue to increase in anticipation of future gains. People would still want to move to desirable locations like big cities, and that increases housing prices too.
So I don't see how exactly this happens without a major, major shift in the housing market - something like "landlording is a revenue-negative activity" or "there is a significant chance of losing your asset value in a fashion that it will definitely never recover within the span of several generations". i.e. nothing like the world we live in.