Only if the housing market is functional and responsive, ie high rents are only driven by "desirability" and so demand can adjust quickly.
The reality is that trends in the rental market are very slow to reverse outside of the collapse of a bubble like in 2008. At the moment it is completely dysfunctional since nearly every city in the Anglophone world is experiencing both a housing shortage and massive speculation in the housing market.
Additionally peoples' housing decisions are much less fluid than their decisions on to say buy a particular product - the entire basis of their financial security is tied up in where they live. Just because rent is lower in a less desirable city that's no help if they can't find a decent job in the less desirable city, or if due to high costs where they already live they don't have the liquidity to finance a major move.
And that's not even getting into the social cost - uprooting yourself from your home can cost you resources that can't easily replaced, such as a social safety net in the form of family and close friends, potentially providing things like free childcare, help in emergencies, rides to work if you can't drive/can't afford a car, interest-free loans etc etc. Because of all this people will stick with unbearable rents until long after they stop being able to comfortably afford them.
Housing responds to market forces, but since the supplier has a far more captive market than most industries the time horizon for changes is very slow. It doesn't really help people to claim that the market will sort itself out so just put up with this misery for 10-20 years - especially when in the past 10 years the market has only become if anything more dysfunctional.