if the air is in contact with the wing, it must push down on it (as the wing will push up against it)
Depends on the pressure actually. I see no reason for claiming a fluid in contact must exert a force in any particular direction.
However, jymil's post with the turning of fluids is a graphic which does show downward displacement - the cambridge video is actual streamlines which show negligible downwards displacement but a significant upwards displacement.
Well, no. You need to look at the video again without a preconceived conclusion.
You cannot beat physical law, and Newtons second law is the only possibility here, unless you wish to claim that it all works by magic or other supernatural forces.