Hi PA
Actually, q as such is dependent on TAS squared and density, only. It has units of pressure and could be considered as kinetic energy per volume unit of air. The more, the greater the aerodynamic forces.
TAS alone won't tell me how the aerodynamic behaviour of my airplane will be if I pull the controlwheel during the take off roll, if it will become airborne or it will strike the runway with the tail, and so many other aspects of airplane handling and performance. It is q which determines those, so I need a measure of q when flying.
The ASI measures q by sensing total pressure and static pressure, as you know. Instead of expressing q in psi or Hpa, the concept of EAS allows us to express q in knots, but these knots are not "real" knots. For each value of q there is only one value of EAS. The same is not true for TAS.
EAS is a convention. It is, by definition, TAS times the square root of relative density. Someone "invented" it (I wonder who) for convenience.
TAS has an influence in manoeuvring stability and other aspects. It is also useful for checking airframe/engine performance in cruise. It is required for navigation, either to calculate GS or wind component. It is a real speed. EAS is not, it's "better" that TAS.