A certain amount of angular pulling got these apart. Thank you again for the advice (breaking these seems like a BAD thing and I'm really not excited about applying a lot of force to plastic parts). I cleaned up the little balls. Assembly seems to be ball first into the handle (opposite to what's shown in the photo) and it definitely is still having trouble breathing after getting those balls moving again (one had a perfect round divot where it had been sitting in the housing for many, many years). Not having the ball in place doesn't allow pressure to be made. Is air supposed to leak in around the leather? That doesn't make sense to me either but I really don't get how air is supposed to get into the pump to enable the pressure build up. Thank you for posting all the photos and your experiences. It really is so very helpful.
Absolutely, yes. Although Primus and other classic brass stoves were equipped with a subtler arrangement to allow air in to be compressed by the pump, Petromax lanterns and others - including this Sigg - rely on air passing between pump cup rim and pump tube on the ‘pull’ stroke of the pump.