Most American watches use a "negative setting" system. The default position is for setting mode, but a sleeve-like spring in the neck of the case allows the stem to "snap" inward and be held in winding mode. The snapping is entirely a function of the case.
On this one, the sleeve was too weak and worn and it was possible, without taking care, to pull the stem all the way out of the neck. None of the parts are broken, just too worn. The sleeve and stem can be tricky to replace. One advantage of the negative setting system is that cases are more or less interchangeable. These interface parts are often custom made or altered to fit a given case/movement combination. Luckily, I found a sleeve that is the same part, but much less worn. This will work a lot better. It's lucky because this sleeve is unusually large. It's an odd type.