I mean, you can get deep into epistemology and argue that it's impossible for anyone to truly know anything, and tautologically all of engineering requires appr

I mean, you can get deep into epistemology and argue that it's impossible for anyone to truly know anything, and tautologically all of engineering requires approximation in some way, but our modern understanding of aerodynamics is absolutely a "first principles" approach (the principles being conservation of mass and momentum in a viscous fluid), even if there are still some aspects of trial-and-error on the aircraft hardware level. Despite the open mathematical problem of the existence and smoothness of Navier-Stokes that you mentioned, the equations are a fantastic tool that have enabled us to make startlingly accurate calculations of lift, drag, stability, and performance, despite our inability to do direct numerical simulation at the Kolmogorov scale on usefully-sized things.