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It's likely the elevator would be set by default to sabbath mode, unless someone specifically pressed the button to go to a specific floor.

Action is (arguably) more important than thought in Judaism.

With that in mind, why are they using the elevator at all? The fact that there is a difference between using and operating is very convenient.

You misunderstand. The key is to get into the room (as opposed to the presumably electronic locks which are usually used) and the elevator is set beforehand.

Not GP but I think the keys were to enter the dorm building without scanning an electronic FOB or ID card, not for the elevator.