Which component's failure would most likely illuminate the green TAD OFF annunciator?

The illumination of the green TAD OFF annunciator indicates that the Trim Aid Device (TAD) is not operating, which is primarily linked to the TRIM AID switch. The TAD is designed to assist the pilot in maintaining control and achieving the desired pitch attitude during certain flight conditions. If there is a failure or an incorrect setting of the TRIM AID switch, this would disable the TAD functionality, thus triggering the TAD OFF light.

In contrast, the other components listed do not directly impact the operation of the TAD system. For instance, the hydraulic pump's failure is associated with other hydraulic functions but not specifically with the TAD. Similarly, while the battery bus is crucial for overall power management in the aircraft, it does not singularly affect the TAD. Additionally, the emergency extension selector is related to landing gear operations, rather than pitch trim functions. Therefore, the TRIM AID switch is the component whose failure directly correlates with the illumination of the TAD OFF annunciator.