It gets complex if you want to fully model things and make it fly as efficiently as possible, but that isn't really in the scope of the question.
Planes go up because they push air down. Simple as that.
You wouldn't explain how swimming works with pressure differentials. You'd just say "you push water backwards and that makes you go fowards". If you start talking about pressure differentials... maybe you're technically correct, but it's a confusing and unnecessarily complex explanation that doesn't give the correct intuitive idea of what is happening.
It is not that simple.
The point is that a flat plane with full flow separations is the minimum necessary physics to explain lift. It would obviously make a terrible wing, and it doesn't explain everything about how real wings are optimised. That's not the point.
In any case, I only said the wing pushes the air down. I didn't say it only uses its bottom surface to push the air down.
