Nice bit of observation by the plods.
I wonder how often policemen and traffic wardens do that sort of check - probably on parked cars - as they are passing on other business. It was said that when tax discs were abolished, it stopped police and traffic wardens doing a cursory check on cars that they were passing to make sure that a) the car had a tax disc rather than a beer label, and b) that the tax disc was for the right car make/reg. Nowadays they have to have reasonable suspicion (rather than just routine check) to do a PNC check to make sure that a car is taxed, insured and (if necessary) MOTed.
My tax disc once fell off the window as I was driving along, and a few seconds later a policeman walked out into the road to stop me for failure to display. I was able to show him the disc on the dashboard, with the holder still wet with my spit where I'd attached it after replacing the old disc with the new one.