Two very relevant examples of when the "amortization" argument won't apply:
- a rising market in which the landlord bought a long time ago. Their purchase price was way below the area's current market value. So they can cover their expenses while you still pay less in rent than what a modern-day mortgage would be.
- a home buying market that's expecting continued pricing appreciation. Homeowners can (and will) rent out property at a loss with the expectation it'll be made up in a higher sale value years down the line. Whether or not this actually happens, of course, depends on their ability to accurately predict the future.
There's also the inverse case of declining markets, when landlords can end up over the heads financially and simply can't charge the rates they need to cover their mortgages, because the entire area has turned south.
There are very real financial risks to landlords. They can't just charge whatever they want. It really depends on the state and future of their market.
In your first example, the landlord is amortizing expenses by charging rent above their ownership costs. It doesn't have to be at or higher than then current new mortgage rates.
The second example does happen. However, if the landlord's bet is wrong, and they have the runway to float the extra expenses for a while, they can simply wait for rents to rise and over decades make it back and begin amortizing then. Or they can sell and hope to make it back.
Property owners have many more financial options with their property than do renters.
Regardless, the point is that a lot of people see the mortgage repayments equalling the rent and don't think about the extra costs of ownership.
You are correct that rent != mortgage and shouldn't be compared too close. Renting is a 100% loss, while mortgage is <100% depending on where you are in your repayment plan -- allowing the owner to recoup some of their housing costs after time.
After the mortgage is paid off, the renter still continues to loose 100% of their housing costs while the owner has only care and maintenance (and taxes) of the property to account for.
Under normal market conditions, borrowing money becomes expensive as monetary supply decreases. However, very active central banking systems the last couple decades has changed the story on this.