As you move the CG back, the elevator becomes more effective. One result is that the previous pitch trim is going to be wrong. Needless to say, it's going to be wrong because the CG move usually makes it wrong, but the amount of deflection is not only wrong for the new CG, but it's even more wrong since it's also more effective. Minor details, but the point is................

As you move the CG back, as the elevator gets more effective, you need less and less. If you don't keep that in mind, you might see things you didn't want to see. Caps are quick to snap with too much elevator.

Also, as you move the CG aft, the effect of speed on trim is lessened. The CG is heading for the neutral point after all.

And a CG becomes a single location only when you get it where the airplane flies the way you want it to. The CG actually has a range of places it can be and the airplane is perfectly safely controlled in pitch. It's positive pitch stability is changed by CG moves, but that's just the recovery aspect of pitch stability. The elevator will simply become more effective or less. And as you move the CG toward the NP the airplane responds to these pitch tests with less and less recovery. But still reacts. It's simply going more and more neutral in pitch recovery. Just as it would go toward neutral roll recovery as the wing is moved up from low wing or down from high wing and dihedral removed. And we do that on purpose, don't we.