One probably still relevant usecase for mechanical state machines are elevators. Original design of push button controled elevators is surprisingly simple and involves mechanically implemented state machine where the carriage itself is part of the controller mechanism (the overall idea is that on each floor there is switch that gets flipped by the passing carriage that signals whether the carriage is above or below that floor), such systems are certainly still used (second generation of automatic elevator controllers implement the same logic "electronicaly" by means of johnson counters built from mechanical relays instead of the distributed electromechanical logic).