The motor on an elevator like Noah is suggesting would have to provide enough force to counteract the entire weight of the elevator + payload + motor + friction, which is at least an order of magnitude more than a traditional elevator.
Not necessarily, no. Put fixed motors on the shaft walls, not in the elevator, and put pinions on the outside walls or corners of the elevator. The only extra weight would be of the elevator itself, less the weight of the hanging cable which elevators today have to move, and less the weight of the braking system, which would now be in the building, not the elevator.
And the much smaller building mounted motors can recuperate some of the energy whenever the elevator is descending.
Because each motor would only have to lift the elevator for a small distance before the next motor takes over, I imagine that higher speeds can also be attained, with less energy expenditure.