Hi Microburst,
As you say, it is almost intuition! I think that in reality everyone will go for an explanation that suits their mind set. For me, a one-time aerodynamicist, I tend to think non-dimensionally, so I seek an explanation in terms of flow angles - AoA and downwash
angle off the trailing edge.
Classical (inviscid) theoretical aerodynamics couldn't explain either lift or drag until a chap called Kutta introduced a hypothesis that the flow should leave the airfoil
at the trailing edge. To make that happen he had to introduce a new variable that he called circulation, so that the flow over the airfoil was a combination of freestream plus a "circular" flow which added velocity on the upper surface and reduced it on the lower. Trouble was, to get such circulation in real life you have to accept a 'starting vortex' when the air first starts to flow over the airfoil. But vortices are viscous phenomena so to make the (inviscid) theory work you have to rely on the existence of viscosity
Anyway, in my non-dimensional world the angle of the flow coming off the TE is only a function of AoA and wing camber. If you grant that, then the downwash
velocity is proportional to freestream velocity as is the mass flow.
From that lift proportional to velocity squared follows directly.