The ASI is nothing more than a differential pressure gauge. It measures the dynamic pressure and gives an airspeed indication that is proportional to that dynamic pressure.
Everything right until here and then you banged the big aviation schools myth that I heard too many instructors quote mistakenly (look, it' s even in the Oxfords according to CCCP) and that induces people like cccp in error.
ASI gives no airspeed indication proportional to dynamic pressure, since the relation between IAS and dynamic pressure is a P = k * V² relation.
The ASI is calibrated to show a given IAS for a given dynamic pressure, but as you say yourself, the relation is not linear, it' s a square.
The ASI needle is calibrated to show a given extension for a given pressure even though it' s not a proportional equation.
But for the interest of the question, all you need to know is that:
L = k * CL * V² = k * CL/4 * (2V)², which is elementary school math.