Evo7
I don't think you're missing anything at all. You've laid out the the dilemma nicely, and I've never seen a "nice" explanation I've liked.
If you're happy with explanations that are fairly mathematical in nature, try the following. Forget pressure and concentrate on velocity for now. The only flows that can be constructed to satisfy the basic equations of incompressible, inviscid fluid mechanics are:
- trivial uniform flows
- vortices
- source-sink dipoles (and higher orders)
Dipoles have the the wrong symmetry (though you need them to match boundary conditions for a wing of non-zero thickness), so the only way to get a downward component of flow at the trailing edge (Kutta condition) is to have a vortex, or collection of vortices.
That leads inevitably to the concept of circulation around the wing -- you can't have a downward component at the trailing edge without an upward component at the leading edge, a downstream component above the aerofoil and an upstream component below it. It's a consequence of the incompressibility.
Is that an "explanation"? At some level I suppose, but it doesn't exactly get the Feynman award for intuitive communication!