No, it doesn't work like that, because it doesn't take the house's capital gain into account.

Given the ability to grow the housing supply, there is typically a fixed return for a property. That return will be split between capital gain and rent. Rent no longer covers the mortgage because it is the total return that determines if it is a good investment. There are also the tax benefits of renting out a house in some locations (maintenance and insurance would be deductible).

When we rented out our house, we were told that the typical return on capital was 8%, where 2-3% would be capital gain, and the rest in rent. We priced our rent at 5% to get a quick rental. Mortgage rates in that location are 6%.

If it is hard to gather a down payment, then rents can exceed mortgages because renters cannot easily swap between the two.

Admittedly, lessons from the SF housing market aren't broadly applicable, but due to rent control, the booming home prices, and bizarre property tax rules, it's much cheaper to rent.

And they aren't always making money. Its about investment.