It's really not. The wing is angled so it pushes the air down. Pushing air down means you are pushing the plane up. A wing can literally be a flat sheet at an angle and it would still fly.
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.
It's both that simple and not. Because it's also true that the wing's shape creates a pressure differential and that's what produce lift. And the pressure differential causes the momentum transfer to the wing, the opposing force to the wing's lift creates the momentum transfer, and pressure difference also causes the change in speed and vice-versa. You can create many correct (and many more incorrect) straightforward stories about the path to lift but in reality cause and effect are not so straightforward and I think it's misleading to go "well this story is the one true simple story".
Sure but it creates a pressure differential by pushing the air down (in most wings). Pressure differentials are an unnecessarily detailed description of what is going on that just confuses people.
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.
Sure. If you're going for a basic 'how does it work', then 'pushing air down' is a good starting point, but you'll really struggle with follow-up questions like 'then why are they that shape?' unless you're willing to go into a bit more detail.
How can you create a 'pressure differential' without deflecting some of the air away? At the end of the day, if the aircraft is moving up, it needs to be throwing something down to counteract gravity. If there is some pressure differential that you can observe, that's nice, but you can't get away from momentum conservation.
The pressure differential is created by the leading edge creating a narrow flow region, which opens to a wider flow region at the trailing edge. This pulls the air at the leading edge across the top of the wing, making it much faster than the air below the wing. This, in turn, creates a low pressure zone.
Air molecules travel in all directions, not just down, so with a pressure differential that means the air molecules below the wing are applying a significant force upward, no longer balanced by the equal pressure usually on the top of the wing. Thus, lift through boyancy. Your question is now about the same as "why does wood float in water"?
The "throwing something down" here comes from the air molecules below the wing hitting the wing upward, then bouncing down.
All the energy to do this comes from the plane's forward momentum, consumed by drag and transformed by the complex fluid dynamics of the air.
Any non-zero angle of attack also pushes air down, of course. And the shape of the wing with the "stickiness" of the air means some more air can be thrown down by the shape of the wing's top edge.
You can't, but you also can't get away from a pressure differential. Those things are linked! That's my main point, arguing over which of these explanations is more correct is arguing over what exactly the shape of an object's silhouette is: it depends on what direction you're looking at it from.
That page is arguing against a straw man. Nobody is claiming that the full dynamics of a wing are exactly that of a flat sheet at an angle (with full flow separation etc).
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.
https://www.youtube.com/watch?v=CT5oMBN5W5M