No idea but holding the bat lower down would move your hands closer to the balance point of the bat.
I also think having more wood in the spine as opposed to the edges might helps pickup. However, getting the weight right is the main consideration. If you can play all your shots with only your top hand it's a start. If you have shove your right hand in to hold the bat through the shot (maybe a little for cut/pull) it's not a good sign.
Yes I think its the weight that comes first. Pickup is quite relative to one's style of batting which ofcourse also depends on the no of grips, where you hold the bat etc etc.
But having said that if we look at Gilchrist he used to have a high grip and could play drive, cut & pull with ease...so there are exceptions to every rule I guess..