I wonder whether the structural integrity of the cabin safety cage is compromised by the door being open (or off), in case of a hard landing or crash
Not too many helicopter types come to mind where the door is part of the structure. No doubt there can be certain panels required for flight. Most door removal restrictions are an aero thing.
If it did it would probably look like the door on a pressurised aircraft!
I have yet to see an F1 with doors!