Also, AFAIK, insects rely on their (flat) flapping wings to create lift by creating vortices.

"Nothing" may be a little too strong -- physics is physics, after all -- but (small) birds and airplanes have Reynolds numbers about 1000x different and different underlying material science.

Airplanes are engineered (with a very small number of exceptions) to be stable during flight, whereas bird flight is inherently unstable, requiring a lot of specialized cognition so birds can fly.

I would think a lot of cognitive ornithologists (who I really hope exist!) would be interested in finding out about this.

At this point, I'm at the complete end of my knowledge of bird flight!