IAS --> (system error corrections) --> CAS --> (compressibility corrections) --> EAS --> (density corrections) --> TAS --> (wind corrections) --> groundspeed.
This is the wrong way. You don't start with IAS! You start with what you need.
You need EAS, not TAS.
EAS is complicated to calculate because you need to know separately Ps and Pt
So you define CAS,being as close as possible to EAS, that can be calculated with Pt-Ps , without having to know separately Pt and Ps.
CAS is easier to calculate. You only need to know Pt-Ps, you do not need Pt and Ps separately
Note that, contrary to a common belief, CAS
does take into account compressibility, but at sea level only. Yet CAS does not take into account density.
Without compressibility, at low speed, CAS would be what I call HAS above
You can check that HAS is a first order approximation of CAS for slow speed.
Then you define IAS as being as close as possible to CAS (the regulation says 3% 5kt), because your pitot and static are not perfectly aligned or perpendicular to the flow, and because the flow is disturbed by the plane.
Genghis said above that IAS is what I call HAS but I don't understand why it should be.