The Navier-Stokes equations merely model fluid flows. Understanding them provides no understanding of the behaviour of such flows. That behaviour is emergent from the interaction of a great many particles.
But that doesn't have any explanatory power at all. If we assume Newton's laws hold, then obviously if there's a force upward on the airfoil then there's a reaction force downward on the air.
It'd be like explaining the combustion engine by saying "the drive shaft from the engine rotates this way, and the reaction force - because the engine is more-or-less rigidly mounted to the frame - is resisted through the suspension by the wheels being in contact with the ground". OK, sure, but I still don't know how the engine actually works.
Added: Or more Newtonish (no action at a distance), there is more upward vertical force contributed by the particles in both cases than downward force.