The ammonia is there because Brasso is sold as a polish that you put dab/pour on a rag and polish your brass doorknob, cannons, whatever, directly. Meaning that the ammonia will not have evaporated and is useful in removing tarnish.
That doesn't happen when you put a bit in a tumbler, it is spread out on the media and the ammonia evaporates. What cleans the brass is the abrasives and detergents in Brasso.
Good question about the ammonia!!! I don't know if the ammonia is useful for cleaning or not, I do however know that ammonia leaves a streak free shin!!!! because it "evaporates" leaving nothing behind. This is why they use ammonia in many other cleaners, especially glass cleaner!! You don't want streaks on you glass anymore than you want streaks on your brass balls do you!!!!!!