I think that the confusion comes from the fact that the CAS is corrected for compressibility, but the correction is only valid at sea level, which makes the CAS a very strange animal.
The seal level only correction has probably been invented because it's the only compressibility correction you can make when only Pt-Ps is available.
My guess is that:
In the old days, there was a speed I could name HAS, historical air speed, which was defined as
HAS = sqrt (2 *(Pt-Ps)/ rho0)
HAS is very close to CAS and EAS at low speed.
Then the need to make corrections for high speed arose.
There is no way you can calculate EAS from HAS only.
So the CAS was invented. The CAS takes into account compressibility, but at sea level only. CAS can be calculated from HAS only, without the necessity to know separately Pt or Ps. CAS is closer to EAS than HAS is.
By the way is there any official or customary name for what I called HAS?