A simpler system might be to have dedicated upward and downward shafts, with change over points in the roof and basement. The cars all queue up to go up or down depending on which shaft they’re in.
The elevator cars couldn’t have cables of course, but they could be driven by electric motors on the cars themselves (powered by induction), driving a gear wheel that runs on a fixed track on the side of the shaft.
I think a Paternoster wouldn't be viable for a number of reasons (mainly ADA/accessibility). Instead you'd need to have each car independently powered and allow them to communicate back to a central planning server to allow cars to stop without stopping all other cars in the lane (and to prevent other cars from bunching up).
The elevator cars couldn’t have cables of course, but they could be driven by electric motors on the cars themselves (powered by induction), driving a gear wheel that runs on a fixed track on the side of the shaft.