It seems the problem with elevators is the cables.
So why not just get rid of the cables? Let them travel on magnetic tracks. Further, have an up shaft and a down shaft and put multiple cars in each. Then switch tracks at the top and bottom. With enough cars, some could bypass stops to account for predominately up or down travel during different parts of the day. You would also need less shafts. Two shafts could do the same work as dozens.
Ask Willy Wonka for plans.