You can actually calculate the rate at which a wing in steady flight changes the downward momentum of "the air". In order to perform this calculation though you need to be very specific about what you mean by "the air". If you decide to consider consider "all" of the air affected by the wing (a sensible approach), you can do this by pushing the boundaries of the "control volume" you use in your analysis out to infinity. This approach gives a somewhat surprising result. Even if the closest boundary of your control volume is an infinite number of spans from the wing, the calculated rate of chage of the air's downward momentum depends on the SHAPE of the control volume. If the front and back (upstream and downstream) faces of the control volume are very tall and skinny, the rate of change of the air's downward momentum is equal to the lift (the pressure footprints on the top/bottom of the control volume contribute no net vertical force in this case). If the front and back faces of the control volume are very short and fat, the rate of change of the air's downward momentum is equal to zero (the pressure footprints on the top/bottom of the control volume contribute a net vertical force equal to the lift in this case). There's no paradox here... the results of either case are entirely consistent with Newton's Laws. In the tall/skinny case the air experiences an unbalanced force from the wing and a corresponding rate of momentum change. In the short/fat case the force exerted by the wing on the air is balanced by an equal and opposite net force exerted by the top/bottom of the control volume.
The idea that a wing's lift is balanced by the rate of downward momentum transfer to the downwash is very appealing (to me anyway). Unfortunately, the reality ain't quite that simple.
