Oleg, I see that my pump gets a lot of use??
There's no magic; it's very easy to explain. First, a pistol grip requires your wrist to cock, which gives you a slightly less sense of direction as far as your hands are concerned. Contrast that with a regular rifle grip, which forces your hand in what I call a "knife holding" wrist cock, which means you are pointing quite a bit more towards your target. Also, AR15's have high sight's with respect to bore axis (one of the reasons I don't like AR15's)
If you compare not so much aimed shooting, but instinctive shooting, an M1A vs. AR15, the M1A has quite the advantage you will find. It points better (again, with the rifle grip, you are like pointing a knife) and the lower sight axis to bore axis helps quite a bit.
You move your arms up quite a bit faster when you have your wrist forward rather then in a pistol grip. Try it empty handed, you will see what I mean. It's which muscles are tight and which aren't.
BTW, been meaning to ask, have you shot your mauser and mine side by side? Is there a big difference in accuracy and easy of handling?