I think I did not explain my problem well enough in my OP.
I am using a 14" M1-Pro MB, plus a brand new Studio Display.
In the first several weeks since I rec'd my M1 MB, Touch ID has been working just the way I wanted on the MB alone [no Display for me yet]:
• if screensaver had started and had been running for a modest [?? exact] time, I could login with Touch ID quickly and reliably, whether using the built in keyboard or a Bluetooth Magic kb with touch ID key;
• if I had been logged out for a while [again, exact time ??] I might be asked [irregularly] for a password;
all good, I'm willing to be asked from time to time to verify.
From the moment I connected my new Studio display, things did not work this way anymore. Every time I was inactive long enough to activate screensaver [yes, I know, that's adjustable by me] I repeatedly [& without exception] got the message:
"Password required to enable touch ID," and I had to enter my password to open my screen.
Eventually, when I simply unplugged the Thunderbolt connection between my M1 MB and the Display, things worked as I wanted. Plugging back in restored the problem - touch ID not available to login.
I stumbled on another thread in this discussion forum where in several posters described the same message and problem. One, and later a few others, reported that if you go to Settings>Touch ID and check the option "Use touch ID sensor for fast user switching"
that this cured the problem - regardless of the fact that they did not have a second registered Account and were certainly not actually switching between accounts.
I checked this option and restarted, and for roughly 1/2 hour things were good; then I began getting the "Password required" message again.
After a couple more plug/unplug cycles of the M1 & Display, things now seem stable in the 'Touch ID works properly' mode.
I've been using Macs since the mid-1980s, and therefore troubleshooting the hardware & software as well. What is this? "Fast user switching somehow affects enabling of touch ID"???
Not an obvious much less plausible relationship to me.
Any ideas??