you could also say that Bernoulli predicts that the airspeed above the wing must be higher than the airspeed below it
Yes, but that is just another part of deflecting the airflow. The deflection causes a pressure difference, and a lower pressure means a higher velocity.
As you say it's all part of the same thing.
There will also be some leakage around the wingtips, so you should expect trailing vortices. Unless you are near the ground...
Or the wing is very (very
) long.