I have my own theories, but I have never experienced a turn coordinator or turn and slip indicator's gyro failing first-hand.
If the gyro isn't spinning, would the instrument behave like a weak accelerometer but on a different plane of movement, or would it just do nothing at all?
For example, would the weight of the motionless gyro of a turn and slip indicator resist movement laterally when the plane is rolled and cause a momentary turn indication from roll or are the centering-springs strong enough to overcome virtually any inertial resistance offered by the weight of the gyro?
Another way of asking this is if you held a turn and slip indicator unit in your hand with the display window facing your palm and you rotated (rolled) the unit side-to-side rapidly in sort of a door knob turning motion, would the turn rate indicator needle tilt at from the center position on the display at all?