> Basically all aerospace engineering undergrads get taught early on that airfoil drag is just zero in 2D incompressible and inviscid flows, [...]

Don't you need some adicional hypothesis, like that all the path integrals of the wind field are zero? Most of the easy part of the analysis of the lift in a wing assumes that air is inviscid (except for the initial generation of the vortex around the wing). [And if the speed is much smaller than the speed of sound, you can approximate that the fluid is incompressible.]

For example see https://www.grc.nasa.gov/www/k-12/airplane/cyl.html