Not if the supply of rental units is increasing. One apartment building can put 100 - 500 units on the market versus perhaps 10 single family houses in that same space.
A good example of this in action right now is Sunnyvale, which has had rising home prices over the last three years but flat rents because the number of rental units coming online has greatly exceeded the available housing inventory.
It would be accurate to say that house rents go up with rising housing prices, but it is not true that all rents go up with rising housing prices. What it does is re-factor the ratio of renters to owners in favor of renters.
I've seen the argument you present used frequently, and I can understand where it is accurate in many markets.
In Seattle, a very similar argument is being made as a cure for the lack of affordable housing -- greater density of construction will lower costs through greater housing supply.
What I see happening in actuality though is that one house that would have fetched ~$900,000 today (and maybe $400,000 just a few years ago, post-recession) is torn down, and four condos put up in the same footprint. Ok - so supply increased, right? Not exactly, because if each unit is priced at ~$700,000, it's still above affordability for most people. So then those units get swallowed up by speculators and investors, the same as what's happened NY, Paris, London, and Vancouver.
I also recall in Seattle there is some building regulation where new apartment buildings get a tax rebate or some similar incentive that lasts 5 years - and then after that the units get converted to condos (or maybe its the other way around). In any event - my point is that there can be a lot of shenanigans that get in the way of the market.
If that were true, the purchase price to rent ratios would all be the same. They aren't.[1]. There are some places in the US (looking at you SF!) where renting is cheaper than buying and vice versa.
[1] https://www.mashvisor.com/blog/best-real-estate-markets-pric...
The market can remain irrational for longer than you can remain solvent.
Additionally, your statement is false under many conditions. It is not close to a proven fact that it is true "eventually," no matter the time horizon.
If you're underwater on a mortgage, it's still the same payment month over month that you were expecting when you took out the mortgage. It'll recover eventually if you hold on. When you're paying rent month to month, that number changes on you as prices rise and you can't just hold on.