Who ever told you that "the bigger the caliber, the bigger the margin or error". has No idea what they are talking about.
No, proficiency, actually marksmanship, is learning to be a good shot. If you are a bad shot. Don't go hunting. Wait until you become a good shot; preferably an excellent shot.
Like I said before; it depends on the animal that you are hunting. Rabbits, squirrels, turkey, and other small game, require a head shot; so as not to ruin the small amount of meat on the body. Deer and other large animals need a well placed heart & lung shot to quickly bring them down,