Equivalent airspeed has absolutely nothing to do with compressibility
Well.. Yes and no. Looking strictly at EAS it deals with non compressible fluids. But as pilots we do not have a EAS gauge. We have a CAS gauge that is flawed due to air getting compressed (and a whole lot of other issues)
Since we have no correct CAS, we cannot obtain the correct TAS when air starts getting compressed. So the formula EAS=TAS*xxxx cannot be used as TAS/CAS is flawed from the beginning.
So if we need to actually know the forces on the airplane from an engineering point of view we need EAS, as this puts the pressure in the non-compressed "domain".
Regarding the design of ASI's, I have no practical experience, but theoretical knowledge books for pilots states that : "The ASI is calibrated to the ideal incompressible flow formula. Because of this a subtractive compressibility factor correction has to be applied"