I was told an unbalanced bow falls the same each and every time whereas a balanced bow could fall in different direction. Who has any 2 cents on this topic?
From a physics standpoint (for any object) it makes some sense. But for archery I think that you may want weight forward to combat the rearward torque of an arrow launching above the support point of your hand. Also I think generally you want the center of gravity below your hand so it's more stable. I guess what I'm saying is for a random object balancing the statement makes sense, but for a bow there are other considerations.
for a balanced bow it should fall forward after the shot, since the bow hand is relaxed as the shot is released the bow will surge forward slightly. At that point the pivot point has changed and the bow should slowly fall forward if balanced correctly.
Add to this that if you shoot with a wrist sling(I do), when the wrist sling catches the bow it applies pressure below the grip.