[wbeaty]

But if air is pushed down, then it must continue moving down (much like with a hovering rocket, or a helicopter.)

But this effect does not appear in typical airfoil flow-diagrams. Instead, the air approaches horizontally, and leaves horizontally. It's a symmetrical pattern, as guaranteed by "circulation theory." In these diagrams, the wing doesn't fling any air downwards. (Heh, the same thing happens with helicopters that fly inside a giant vertical pipe, where the blade-tips slide along the inside surface of the pipe.)