You don't understand enough about real estate, or finances in general, to make statements about mortgages or rent.
My mortgage is about $1,200 a month. Could I rent my 2,300 sq ft home for that? Perhaps, but unlikely. And the landlord could kick me out once my lease expired. And he gets to say what I do with the property. And he owns any improvements free and clear, unless we make written agreements.
Besides appreciation, as you pay your mortgage you gain equity. Equity is the difference between the property's value and your loan balance. Even if it doesn't go up in value at all, you "own" that equity. When you rent a property for 10 years and move, you get nothing (except perhaps a little interest on your deposit). When you sell your home, you get back your equity.
Should people not be buying extravagant homes and calling them an "investment"? Indeed, they're a liability, considering the money you have to put into them all the time. But sweeping statements, like "rent is cheaper", doesn't show the whole picture. Right now, rents are still pretty high, so those who can find a way to make deals on property are winning.