I fixed it. Not OEM fixed, but "works for me" fixed.
Get some Nylon webbing, cut it to an appropriate length, and melt the edges so it doesn't fray. Then drill a hole through the strap and case and pop-rivet the webbing-turned hinge into place. My straps were soft and kept grabbing the drill bit. I was successful drilling through them on wood. For extremely recalcitrant straps I would have melted through them with a soldering iron.
It works! Now I have to go find all my sockets. Murphy's law dictates the most useful sockets will be least accessible spot. If anyone needs me I'll be magnet fishing under my workbench behind the pile of shingles in the sawdust and spiderwebs. :/