I've been down with a bug most of last week and through the weekend, so I haven't accomplished much. I did feel up to running a bunch of gunked-up shifters through a zillion passes in the ultrasonic cleaner, and they seem to all be clicking happily, even after the "Below Zero Test." Considering that some of them wouldn't move at all prior to cleaning, let alone give even one click, I suppose it's an accomplishment.
I think what I will do is collect a few more photos of the process and write it up in the Bicycle Mechanics forum for posterity, to share a method that seems to work for a much longer time than the popular "flush and rush" method seen in videos and other discussions.