Hmm .. thinking it through .. the pressure is not always equal (turns, etc) so what is the importance of balancing the elevator? (assuming there is no gross out of balance situation)
The primary purpose is to decouple the elevator from motion (torsion or bending or both) of the horizontal stabilizer. Consider this basic example.
Case A has the CG of the elevator well aft of the hinge line. Case B has the elevator's CG located on the hinge line.
Cause a sharp vertical displacement of the horizontal stabilizer.
With A, the hinge line will displace with the HS, but because the elevator CG is aft of the hinge line, inertia will cause the elevator to lag behind the motion. The deflected elevator applies a force to the hinge line, driving it in the same direction as the original motion.
With B, the hinge line and CG will be displaced equally. There is no
relative motion of the HS and elevator, thus no new force applied to the hinge line.
Now consider the same A and B, but this time instead of a single vertical displacement, make the HS displacement repeating, driven up and down and up and down by aerodynamic forces.
With A, the elevator will always lag behind the HS motion. The amplitude of the relative motions may stabilize, or it may be divergent, gaining amplitude with every cycle until some component fails due to structural overload.
With B, the elevator remains in sync with the HS stabilizer displacement.
The complete picture is much more complex, and best understood with math beyond my capabilities. We have some very bright people here who can offer a more in depth explanation.