> Of course, on paper it's a terrible idea to pay off a mortgage early (assuming a reasonably low interest rate).

Why? The longer my mortgage lasts, the more interests I'm paying (no matter how low they are). If I could pay the entirely of my mortgage now, I would because it will be cheaper.

I don't think rich people (the really rich ones) pay their houses valued in 1 million, via mortgages. They pay them upfront (because they have the money). Why on earth would they apply for a mortage that lasts 10-20 years? They would end up paying 1 million + interests.