I guess you can also see it from conservation of momentum. Gravity acts on the plane, so each second there is a certain amount of downwards momentum added to the system. But the plane is not in fact moving downwards, so the all the downwards momentum has to end up in the air, so it has to move downwards. The net "lift" force is equal to the rate at which momentum is added to air.

If I understand it correctly, you can calculate the lift on the wing either in terms of the pressure on the two sides, or in terms of the downwards acceleration of air--it should be two equivalent views giving the same result. Bernoulli's principle gives a way to calculate the pressures if you only know the velocity of the air (this is a typical situation in a wind tunnel, where you make movies of moving smoke puffs), but it's not a separate effect from the air motion.