Coggan’s TSS, as I understand it’s genesis, is an adaptation of Bannister’s TrImp. TrImp was a heart rate metric & Coggan was looking for something similar but power based.
TrImp is: Time * DeltaHR * IntensityWeighting
Where DeltaHR is heart rate expressed as the increment of heart rate above some reserve heart rate.
So I think what happened when Coggan tried to transfer this from heartrate to power…he used Intensity Factor to replace IntensityWeighting. And he used NP/FTP to replace DeltaHR. And reducing the TSS formula to IF^2 was a quirk of that construction that became obvious later.
Really, Coggan’s attempt to weight TSS by IF probably wasn’t as successful as he would have hoped. TSS inherently OVERestimates lower intensity training stress and UNDERestimates higher intensity training stress. TrImp was designed to do this much less so…that was the specific intent of the TrImp IntensityWeighting term.
Also, Bannister honed his formula with a body of empirical data. I’m not convinced that Coggan did but what TSS is, is what TSS is. Maybe the best thing it has going for it is ubiquity & wide acceptance.