This will result in a correspondingly higher ground speed in all phases of flight.
Nope.
GS accounts for the effect of wind. A headwind will result in a GS that is lower than TAS while a tailwind will result in a GS that is higher than TAS.
Plus, you are correct that TAS is IAS corrected for all of that, but each of the corrections have specific terms - Calibrated Airspeed (CAS) and Equivalent Airspeed (EAS). So while the method is correct, it is technically incorrect to say that TAS is IAS corrected for all of that.
The technically correct method is to convert IAS to CAS to EAS to TAS then adjusted for wind to GS - although, granted you can get away with IAS to TAS for most general aviation and commuter aircraft and that is how many a POM or AFM is written.
In saying all of that - when flying under perfect ISA conditions, CAS is the same as EAS and TAS and only in nil wind conditions is it the same as GS.