Can a helicopter have a fully mechanic "elevator"?
Using a building elevator would cut the payload of the helicopter in an unacceptable way. Also, none of the safety features of a building elevator could be implemented, and in the end you will have a cage lowered on a rope. Which is quite similar to existing winch systems.



Elevator components (source)

In addition to the motor and the pulleys, a building elevator has a car which runs along guide rails, and a counterweight to compensate for the heavy car. Adding all this to a helicopter would require a much bigger machine just for carrying the elevator components around. Note that using guide rails would dictate a specific hover height - no helicopter pilot would agree to have his options restricted that severely.

Lowering anything in the downwash of a helicopter rotor will cause it to swing and rotate, and putting the occupants into a cage will cut their ability to stabilize themselves or the cage. Once the cage is raised all the way up, it needs to be oriented correctly for easy egress. A rescuer on a rope can easily stabilize himself by grabbing handles on the outside of the helicopter. Sitting in a cage, he can only watch helplessly when the cage arrives in the wrong orientation.

There are many scenarios where rescue helicopters cannot land, like over water or over a forest. In those situations a winch is used to hoist one rescuer down to the ground, where he/she can pick up the injured and secure them to their harness. Both of them are then raised up and taken on board. Only when the patient must not move (think fractured leg), a platform is used to secure the patient. See below for a simulator installation, and note that still a rescuer is needed to stabilize the platform.



Helicopter rescue hoist trainer (source)

If the ground allows, landing the helicopter is simpler, makes much less noise and makes it immune to changing winds. Also, lowering the rescuer first and landing the helicopter will take a similar amount of time, so no substantial time savings can be expected.



AgustaWestland AW139 helicopter on SAR exercise (source)