Ok, after experiencing this as an intermittent issue for a few months, today it affected three Xbox Ones, seven controllers, and four headsets. I tried turning off instant-on, reprogramming controllers to other consoles, changing all audio settings, with all different combinations of equipment.
The ONLY thing that worked at all was to insert the 3.5mm jack into the controller VERY slowly. I pushed it in until the Xbox showed the headset connection and then pushed all the way in. This low-tech hack idiocy actually seems to work as a temp fix.