To update this saga...
To replace the failing one, I bought a new rotary switch. "It's intended for a 3-way circuit, but it'll work with ordinary circuits too." Turned out "work" meant it'll flip the light on and off, but not dim. If I didn't want dimability (word?), I'd have purchased a common, standard wall switch.
OK, took that one out, replaced it with a new Lutron non-3-way. It does as desired except it lacks illumination of the knob. This is for bathroom use, and at night the tiny light eases finding the switch.
OK, took that one out, replaced it with a new GE illuminated non-3-way. We're getting closer! This one does what it says, what they don't tell you is it makes a LOUD CLICK when pushed on and off, less than ideal for sleepytime use. Also, there's are two relatively minor user interface issues that the former switch handled better. When clicking the knob to turn the light on, this new switch does not send the current to the lamp until you release the knob. That means you can't adjust the brightness without pressing, removing your hand, then reaching back to turn the knob. Additionally the knob is a smooth circle that lacks tactile feedback, so in the dark you can't tell what brightness level will come on initially. The knob is plastic, so I can solve the feedback issue by filing a notch into it.
The engineer in me wonders who designs these things, and if they test them in real situations.