Comment by Gibbon1

My dad who worked on wind tunnels just flat said you can either integrate the pressure over the surface of the wing or the momentum change as the air passes over to derive the amount of lift.

Both give exactly the same results and are convertible mathematically.

For wind tunnel work it was easier to measure pressures.

I'm with you I don't think the standard hand wavy explanation gives you the ability to attack the problem mathematically. So it's basically wrong.