Just to point out the obvious problems of having the elevator door start open: 1) you now have a common case where the door starts to close as someone is just

Just to point out the obvious problems of having the elevator door start open: 1) you now have a common case where the door starts to close as someone is just about to walk in, startling and/or annoying them, and 2) not only does closing the door first increase the time to get to any other floor, but the elevator doesn't even know whether the door close will be successful (it could be held or otherwise obstructed) -- leading to even more delays on another floor.