Not necessarily. While very unlikely, the jackscrew nut or some other attached component can fail catastrophically, instantly releasing the stab to full nose-down control.
So far we only had cases where those actuators failed while they were moved, so it is very unlikely that this would not be recorded on the FDR. The position pickup of the THS is independent from the actuator itself, but of course, if the event happens very rapidly, the data will not make it from the stab to the cockpit, back to the FDR and into static memory. And of course an event in the tail may knock out the FDR instantly, as it is installed exactly there.
The relatively low damage to the tail section makes it not very likely that something happened exectly there.