This is a bit simplified and not strictly accurate but as a rough explanation. Aerodynamicists will have conniptions but here goes...
Generally for flying the aircraft, IAS (as measured by the ASI) is primary for the pilot. It measures how the aircraft 'feels' the air.
As air density decreases, the aerodynamic force provided by the movement of the air over the craft decreases. IAS will reflect what the air "feels like" to the aircraft. So for the same IAS but different altitudes, the aircraft seems to fly the same way (meanwhile the TAS will change).
Vne is a bit different.
Vne depends not just on the 'striking force' the air imparts on the aircraft (if it did Vne would remain at a constant IAS regardless of altitude).
It also depends on the actual speed of the aircraft through the air (even at lower densities).
eg aerodynamic "flutter" is more like a resonance or vibration rather than the result of the direct force of air striking the airframe. It is more related to actual airspeed and so as we fly higher our IAS may remain below the lower altitude Vne but if our TAS is high enough to cause flutter we may run into dramas. So the IAS for Vne will change as we go higher.
Like I say this is very simplified and not 100% accurate (I am not an aeronautical engineer) but hope it gives a rough idea of why the ASI is used so prominently.
NB there is also CAS (calibrated Airspeed), EAS, (equivalent airspeed) as well!
.