Why don’t airliners use stabilizer movement to assist with large pitch control inputs?
The elevators hydraulic actuators are quick response devices, the THS is screw operated and therefore is slower to response. 
Therefore it’s normal to use the elevators for short term action and theTHS for long term action.

For safety you can’t rely only on elevators or only on THS. One system should be able to neutralize the other in case of failure.