Got another flat -- second in four rides -- and the gizmo helped a little. I did manage to fix the flat the first time, and a group of fellow riders showed up just as I was finishing and offered moral support.

Getting back I went for a good inspection of what might have happened, and it seems the rim tape was a tiny bit too narrow for the rims, so I suspect that -- over time and distance -- wore a little hole on the inside of the tube. So (with help from the spouse), I changed out the rim tape, and then put the new Gatorskin tire on for good measure. Now I just have to hope for no more flats, because there is no way I will ever get that sucker off and on again.