Why don’t airliners use stabilizer movement to assist with large pitch control inputs? Most large airliners not only have elevators for pitch control, but also moveable horizontal stabilizers.  The elevators are used for primary pitch control, with the stabilizers being used to trim the aircraft, and, for some airliners, as an alternate method of emergency pitch control in the event of a loss of elevator control (due to, for instance, a failure of the hydraulic systems powering the elevators, an elevator actuator hardover, or the physical separation of one or both elevators from the aircraft).  Due to their much larger size, the stabilizers have a vastly greater maximum control authority than the elevators (which is why large modern airliners can be trimmed throughout very wide center-of-mass longitudinal-position ranges without having to defair the elevators, and why a stabilizer trim failure is generally somewhat inconvenient); however, despite this, few, if any, large airliners use the stabilizers to assist the elevators when making large longitudinal control inputs (for instance, when taking evasive action to avoid a MAC, or when recovering the aircraft from an upset).

Why is this?  Airliners can and do use secondary flight controls to augment the primaries in other situations; with roll control, for instance, small inputs are handled by the ailerons alone on most aircraft, while the spoilerons1 jump in to help the ailerons with larger changes.  So why aren’t the horizontal stabilizers called on to assist the elevators when large, sudden changes in pitch are necessary?



1: A spoileron is the technical term for a spoiler that is also used for roll control in flight (these are usually the same spoilers as those that can be extended symmetrically in flight to slow the aircraft).  Essentially all large aircraft, and many smaller aircraft, have them to help with large roll inputs; a few aircraft, such as the Mitsubishi MU-2, have only spoilerons, with no conventional ailerons.
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.