TouchID stopped working

TouchID on my 2021 iMac X1 worked perfectly until I upgraded to Monterey. Now it refuses to work.

I (possibly wrongly) assumed, that I needed to register a new fingerprint for some reason. So I tried to add one using "Add fingerprint" in the settings. I entered my password manually. All I get is an instant response "Failed: Unable to complete Touch ID enrolment. Please go back and try again." I'm not even led through a process of trying to register the fingerprint. I have tried several times, I even took the option of removing the old fingerprint. Still no go.

Oh dear, this is the sort of thing I heavily overspent my pensioner's budget on this machine to avoid, my sanity having finally reached breaking point after years Windows updates regularly messing everything up and wasting far too much of my time. So far it has behaved itself perfectly and been a joy to use.