The holes had gotten too big for sheet metal screws to hold, based on the size of the holes in the mounting plate that attaches to the door. I tried to use Well Nuts. A rubber bushing with a brass nut in the end. Put the bushing in the hole and when you tighten down the rubber expands and holds whatever you are attaching. These work and won't rust but the mirror still has a little movement to it.
So I installed well nuts. Pretty sure they are aluminum but they hold tight. Yes, steel and aluminum can do some weird stuff over time. Just have to see?