I have been in elevators in which it does do something. I've timed the difference. This foul rumour must die.

They effectively don’t do anything in most elevators in the US during normal operation. The ADA requires elevator doors remain open at least 3-seconds. Usually, people-moving elevators are most efficient when doors close as quickly as possible, so they start closing exactly at 3 seconds. I’ve used elevators with less common use cases — huge ones in hospitals, freight elevators, hotel service elevators — that might be configured to stay open longer than the 3 second minimum, assuming people will push the door close button as soon as they’re ready.