I poked some parts with a toothpick a bit and I think I might know what the problem is. The chrono cam (2) doesn't have the range of movement to actually put the chrono gear in contact with the central hand's gear. It gets close, but the end of the cam close to the pillar (1) actually comes in contact with the edge of the screw before the gap between the chrono gear and the central gear (3) closes fully and the gears engage.

I think maybe if I just turn the screw on that pillar a little clockwise, it might let the cam move far enough to get the gears to engage. Not sure how much room for error there is for something like this, but maybe whoever did the service last time didn't care about the position of that screw since the chrono was wired away from that section in any case. Only problem is, even my smallest flathead doesn't fit into these heads. I don't want to use pliers on something this delicate, though. Maybe a knife... I'll wait a while first and see if anyone here wants to weigh in on how I'm about to catastrophically destroy my apparently valuable watch.