As it happens, I had already fixed the same problem on the *other* side of the door. Today, I finally had time to look in the Kitchen Junk Drawer (Thanks, Wulf!) where I found the little packet of screws and the little packet of nuts that I had used in the previous repair. I removed one of each from the packet, got a wire cutter to trim the ends of the rivet so it would come out of the slot, and -- after a bit of screwing around -- the door was securely attached again.
This is not a major thing, but I am happy to take such victories as are available for the taking. :)