They are installed like a pop rivet.  Thread a long screw into it with a washer and a hex nut under the head of the screw.  Install screw so threads are fully engaged in the insert.  Washer will be next to the head of the insert.  Grease the washer.  Tighten the nut while holding the head of the screw stationary.  That will deform the part inside the door so it crushes against the sheetmetal holding it in place.

Or, get the special tool designed for installing those things, like a large pop river tool.

Luckily you have direct experience with how well those things work and hold up.  I prefer a piece of 1/8" thick steel under the door skin with both holes drilled and tapped to match the mirror screws.  Apply RTV to the steel and hold it in place and install the mirror.  The RTV will hold that steel piece in place if you ever have to remove the mirror again.  The steel piece will not spin, nor will it fall out, unlike the original type insert.