You are all hinting around it but not quite hitting it.
Aerodynamics 101.
An airplane flys because of the wing. It is designed so that air takes longer to go over the top of the wing than the bottom and that creates negative air pressure or lift.
OK that is the very basics of it. Now comes symetrical wings. Angle of attack comes in to play more. As you angle the wing in relation to airflow you present a larger surface area on top as compared to bottom resulting in the same negative air pressure or lift.
Inverted flight? same thing and that is why flat bottom wings have more lift and require a higher angle of attack when inverted.
It all boils down to air traveling farther on top (or bottom when inverted) to create lift.