Hi Owain
something like that, but as I say it is kind of an intuition. We can investigate this further.
The first part, the increased mass flow, I think it is clear: the faster the airspeed, the more mass of air is pushed, the more lift is generated. The other part, the increased acceleration due to increased airspeed, that is another matter, but seems logical: the stronger the push by the wing, the steeper the pressure gradients around the wing, the faster the air will flow as a result and the stronger the upwash and downwash effect.
that would explain speed squared
The truth is that the empirical approach is clearly the best.