Just to clarify - I never said the method of pushing the gun away from your body before you bring it to your shoulder was wrong; I said that if you have to do that to mount your gun then the LOP of that gun is too long.

Even if the LOP is correct, that method is not one that I prefer because it requires an additional movement to obtain a proper gun mount. I believe that the more proficient and efficient you are with your gun mount the more targets (or birds) you will hit.
In some disciplines (live pigeon, particularly Columbaire) I don't think you have time for anything but the most efficient gun mount. If someone says they shoot better doing it the other way then that's what they should do. The bottom line to any shooting system is to answer this question - By doing this am I shooting better or not?

I'll throw two cents in on pitch while I'm here. Pitch, in and of itself, has no efect on where a gun shoots. I shoot with some guys that can break more targets shooting from the hip that I can with my best gun mount. Since the gun is not on their shoulder what possible difference can the pitch make? (i.e.,the Tom Knapp example given earlier).

Having said that - it all comes back to the gun mount. If the pitch isn't right you are more likely to improperly mount your gun. If you don't have enough pitch you are more likely to catch the heel of the stock on your shoulder, and if not corrected, you will shoot high. Reverse is true with too much pitch, you may catch the toe and shoot low. Again, this really doesn't apply to to disciplines where a premounted gun is allowed. It is most applicable in disciplines where you have a fraction of a second to acquire the target, mount your gun and kill it.