I don't think this is the case. Maybe it's a placebo button in some cases? But I've definitely ridden in elevators where the "close door" button has nearly immediate effect (usually older ones). The button never overrides the safety bumpers that reopen the doors, they just bring about the natural closing sooner.
Elevators have a “manual mode”. This is mandatory by law to exist for firefighters/emergency workers, but can be used for other purposes (eg when moving into a high rise apartment, you reserve the elevator for a few hours to move furniture), in commercial buildings sometimes the cleaning staff will use it to hold the elevator and their equipment on their floor as they go floor by floor.
Manual mode does override the sensor that checks if something is in the way, but doesn’t override the pressure sensor that responds if the door hits something when closing. You usually have to hold the close button while it beeps to close the door.
The buttons exist for manual mode. If they work elsewhere, that’s lucky. When I lived in a high rise, it took ~20s for my elevator doors to close. If you were the last one in, you might hit the button and see the doors close immediately, but only because of you timed it right, not because the button did anything during day to day usage. I’ve timed it, and my Otis elevator didn’t close faster due to the button.
The “open door” button obviously works most places.