How does a failed turn and slip indicator behave if the gyro isn't spinning at all? 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?
It depends on how it fails, there are lots of possibilities on what caused it to stop spinning and they can all lead to different outcomes. In this instance it just kind of bobbed around as if it were free floating. But if something is jamming the mechanics it may simply stop moving altogether so it really just depends on how it fails.