The outlet probably turns outward, so it’s not just vertical movement.

Ah, I did not think of that. So I need to scrap the idea of a rigid pipe going into a rubber bushing. I guess an accordian pipe would be the viable cheap hack, apart from some way to add support just around the drain.

Maybe there was some kind of metal support that got left out during installation?

The toilet fell through the floor at one point because (I’m told) the house flippers neglected to use backer board wherever they layed tiles (kitchen and bathroom).