Hang on a moment, you are forgetting inertia. Just because this balance point is called the centre of gravity, dpoes not mean it is only related to the earth's gravitational pull. The CG position WILL determine the behaviour and stability of the aircraft in pitch as well as yaw. Having said that, yaw performance is hardly affected because the CG range will never come close to producing more side area infront of the CG than behind it, or even affecting the ration to a significant degree.
Lift acts at the centre of pressure (CP), not the CG. The relative position of the CG and CP will produce a pitch over effect which is balanced by the negative angle of the horizontal stab. These rules change with FBW aircraft, especially the unstable ones.
Yaw is different in that it is not counteracting a constant force (gravity), but maintaining directional stability. Now, if Milt's theory was correct, and CG played no part in the equation, why isn't the V/stab in the centre of the plane? Having the V/stab in the tail ensures that the lateral CP is well behind the CG (much more than the vertical, or lift CP). FlyingForFun is correct when he states that the distance of the V/stab from the CG determines its moment arm and therefore effectiveness in terms of yaw stability and authority. Taking the 747sp as an example; when the fuselage was shortened, the size of the V/stab had to be increased to maintain its effectiveness because of the shorter distance from the CG and reduced moment arm.
There is no question that changuing the CG will affect the effectiveness of the V/stab in terms of stability and rudder in terms of authority. However, because of the large distance of the lateral CP from the CG, this effect will, in practice, be negligable compared to other factors. I seriously doubt that any pilot can detect the difference in yaw stability and authority when the CG moves within its permitted range on most aircraft, especially large commercial transports. The differences in question can be natural yaw oscillation period, rudder authority, susceptibility to PIO and extreme attitude recovery amongst others. Certification is supposed to ensure that these differences remain negligable as long as the CG remains within the envelope. Any aircraft that does demonstrate significant differences in yaw stability/authority would be, IMO, a bad design.
Where the CG can have an effect on yaw stability/authority is in marginal control situations, such as spin entry and recovery. While rearward CG condition does have serious implications on pitch authority and therefore makes spin entry much easier and recovery much harder, this is mainly due to pitch characteristics, but, is also true because of the reduced effectiveness/authority of the V/stab. That is why certification relies on comprehensive flight testing throughout the CG range. My only concern here is that both large passenger jet manufacturers rely heavily on computer simulations and extrapolation of flight test conditions to determine the performance in extreme and marginal conditions.
I haven't read any operational text books which conclude that CG is a "significant" factor wrt yaw stability, but that is because the location and size of the V/stab has been designed to perform as advertised as long as the CG stays within the permitted envelope. However, at the design stage, the exact size of the V/stab is mostly dictated by its location relative to the CG.
So, to say that CG has no effect on yaw stability is technically incorrect. However, in practise, it's hardly worth worrying about compared to the effects that CG has on pitch.