Except there is no point in paying interest to the bank just in order to get a deduction. There are other ways to shield that money and still keep it. A person paying off their mortgage could turn around and fully fund a 401k plan and thereby reduce his or her tax liability.
A person paying off their mortgage The typical after-tax 'cost' of a mortgage today is something like 4%. If someone had several hundred thousand dollars of cash to pay off their mortgage, it would be better for them to just put that money in something like a 401K. Once it's parked into "the house", you don't have access to it.