Comment by ActorNightly

To be more specific,

You can't escape momentum exchange. To generate an upward force, the airplane must exert a downward force on the air molecules.

An airfoil does this more efficiently than a flat plate, essentially using the top shape to create a low pressure area that sucks the air over the top downwards, imparting the downwards momentum, along with deflecting the air downward on the bottom surface. A flat plate pitched upwards "stalls" the air on the top surface, because the air has to travel forward some to fill the gap by the plate moving forward - so this creates a lot of drag as the plate is imparting more forward momentum on the air.

The issue is that to analyze lift using momentum, you have to do statisitcal math on a grid of space around the airfoil, which is super complex. So instead, we use concept of pressure with static and dynamic pressure differences creating lift, because it makes sense to most people learning this, which then all gets rolled up into a plot of lift coefficient vs angle of attack.

And as you dive deeper, you learn more analysis tools. For example, there is also another way to analyze performance of an airfoil more accurately, which is called vorticity. If you subtract the average velocity of the airflow around an airfoil, the vector field becomes a circle. In vector math, the total curl of the vector field is directly correlated to the effective lift an airfoil can produce. This method accounts for any shape of the airfoil.

But under the hood its all momentum.