Actually, it appears that the Cm gives the answer for what additional work the horizontal tail has added to it's primary job, maintaing the pitch of the airplane. The amount and sign of the Cm describe this extra effort. But the tail's task is to keep the pitch constant until we throw elevator into the job. And where the CP is would affect that task.

Tail moments are usually figured considering the convention that the distance from wing to tail is from one fixed location in one to a similar location in the other. 25% MAC is often used. But CP locations are also used for finer predictions. Which of course are also far more tedious to figure. And tail moments are used in pitch stability estimation because they are a primary player.

Speed a Speedy Bee up and what happens? The AOA is lessened. The CP moves aft. The tail moment is effectively shortened. The tail needs more area or AOA to do an increasingly difficult job of holding the airplane's nose up. It won't find any more area and it doesn't control the AOA, it's own or the airplane's. The wing does. And the Bee speeds up a bit more........