Most visitors online was 1660 , on 12 Dec 2020
Your story is right. One or two (depending on the movement) of the pins on the pinion are longer than the rest, and as the pinion turns, it advances the count wheel. That missing pinion gear you are asking about is not missing. The solder on the end of the lantern pinion is where the longer pin was. Someone replaced it with the wrong pin.Well...........that's my story.......and I'm s....s....stickin' to it .
Your story is correct so stick to it. Here are a couple pictures from a similar Ansonia movement.Didn't they use one long trundle on that lantern pinion ? That's why the count wheel has saw teeth form . Solder on the lantern looks like someone has been there before .