So it turns out that the phone handset is an actual antique, and there are no smarts in the cable! The mini stereo jack is wired straight through to the four pins on the RJ11 on the handset in the order "ring, middle, tip, ring". So it's trivially easy to wire up one of these yourself if you have a handset already.

This means that the "answer/hangup" buttons on wired cell phone headsets must work by inline signaling: my guess is that they just send a momentary break.