The third law could explain lift, but it does not explain why the pressure differential would exist, and thus cannot account for the extra lift due to that.
There is no "extra lift" due to the pressure differential. The pressure differential IS the net force that changes the motion of the air. You can't change the motion of the air without a pressure differential, and you can't have a pressure differential without changing the motion of the air.
There is no "extra lift" due to the pressure differential. The pressure differential IS the net force that changes the motion of the air. You can't change the motion of the air without a pressure differential, and you can't have a pressure differential without changing the motion of the air.