the faster the air is, the more accelerated it is
Why would that be?
Well, this one I must confess I am making an assumption. That assumption is that for a given set of conditions, the speed imparted to the air is a "given percentage". So the faster the air stream is, for that same "percentage" means that the air is imparted a higher speed. Something like: if airspeed is 100 kt, the air will be accelerated 10 kt (a 10%). If the airspeed is 200 kt, then the air will be accelerated 20 kt. Double acceleration, double force.
If I am not right, then there must be another reason why increasing speed increases lift, aside from more air being "processed" in a given instant. Otherwise speed wouldn't go squared.
Ah, I see what you are getting at. It was the use of " more accelerated" that fooled me - not thinking straight
As you say, if you double the airspeed then the mass flow rate over the wing doubles and the downwash velocity doubles (at fixed AoA). The wing lift then increases by mass flow rate* downwash velocity i.e. by a factor of 4 (speed squared).