The Bernoulli effect as a separate entity is really a result of (over)simplification, but it's not wrong. You need to solve the Navier-Stokes equations for the flow around the wing, but there are many ways to simplify this - from CFD at different resolutions, via panel methods and potential theory, to just conservation of energy (which is the Bernoulli equation). So it gets popularized because it's the most simplified model.
To give an analogy, you can think of all CPUs as a von Neumann architecture. But the reality is that you have a hugely complicated thing with stacks, multiple cache levels, branch predictors, specex, yada yada.
On the very fundamental level, wings make air go down, and then airplane goes up. Just like you say. By using a curved airfoil instead of a flat plate, you can create more circulation in the flow, and then because of the way fluids flow you can get more lift and less drag.
That said, I recall reading somewhere that it's a combination of effects, and the Bernoulli effect contributes, among many others. Never heard an explanation that left me completely satisfied, though. The one about deflecting air down was the one that always made sense to me even as a kid, but I can't believe that would be the only explanation - there has to be a good reason that gave rise to the Bernoulli effect as the popular explanation.
And you can tell that effect makes some sense of you hold a sheet of paper and blow air over it - it will rise. So any difference in air speed has to contribute.