So the 18k for the mortgage just disappear when someone else owns the place? Do you even listen? Unless you're renting from a corporation (in which case the rent is not going to be cheaper anyway, they'll just make more profit) the landlord is paying that same mortgage (maybe only 17k since they have a longer history with the bank). You're not getting out of this cheaper. Any cost you might have with a house the landlord has as well, at best they get a better credit from the bank but overall the difference is so miniscule it doesn't balance out the cut they add for themselves onto the rent.
Regarding your second paragraph see my list of reasons why you might legitimately rent. If the saved time is worth it for you then that is absolutely valid but don't delude yourself into thinking it's cheaper.
The point is that it doesn't matter if the landlord pays the mortgage or you pay it it is still there. And if the landlord pays it you can be damn certain it's gonna end up in your rent calculation.