But every airfoil has an equilibrium angle of attack (not always stable with velocity) where it generates zero lift. The chordal angle of attack is for convenience because it depends only on airfoil geometry and not ambient velocity, but it isn't a fundamental physical property of the airfoil.

If we treat the angle where zero lift is generated as the base angle for an airfoil, then all airfoils generate lift depending on their angle relative to that, including a flat plane. As the GP says, other properties are the dominant factor in airfoil geometry.

When introducing airfoils I think it is more useful to start from a plane than a traditional airfoil shape; the math and intuitions are much clearer from there.

And with steady level flight symmetrical airfoils are flown at an angle, a cambered airfoil shape being flown at 0 degrees angle of attack vs its chord line would be an unusual coincidence. Wings are mounted at a small angle relative to the direction of thrust and what one would define as a flat line on the fuselage.