As a child, I quickly outgrew the airfoil explanation when I realized this.
In a normally flying airplane, the wing compresses and pushes an amount of air under its wing. But there is actually even greater amount of air sucked down by the region of underpressure created above the wing and by the laminar flow directing it downward. Here, the drawing at the top of the page makes it clear: http://www.amasci.com/wing/airfoil.html
When you have a stall condition, what happens is that the air below the wing is still being compressed and directed downwards, but the air above the wing becomes turbulent and "unsticks" from the surface of the wing. Rather than being nicely directed downward, it just dissipates a lot of energy in turbulent motion that is not directed in any particular direction.
This turbulent air not only ceases to provide lift, it also prevents the air from below the wing to be directed downwards efficiently.
The main job of an airfoil isn't to create a pressure difference, it is to create conditions for the air to be laminar at as wide range of speeds and angles of attack as possible to make the plane nicely behaving and possible to takeoff and land. It is super critical for landing as you need to have higher the angle of attack the slower you fly and all planes essentially are driven as close to stall as possible during landing. Similar happens at high altitudes and high speeds, but for a bit different reason (read up on "coffin corner" if you are interested in that sort of thing).
The reason wings produce significant lift anyway is that they deflect air far beyond their surface. Air several metres away from the wing is also deflected downward, even though it doesn't actually hit the wing itself.
So yes, Newton's third law is involved, but in a "spooky action at a distance" form, where the wing somehow manages to deflect a bunch of air it doesn't even touch!
Both give exactly the same results and are convertible mathematically.
For wind tunnel work it was easier to measure pressures.
I'm with you I don't think the standard hand wavy explanation gives you the ability to attack the problem mathematically. So it's basically wrong.