Heat is a vey emotive subject, people who don't have it tell you that's it's not needed, and give the wool jumper example as why you should not use high heat on wool..... "You would never wash your wool jumper in hot water"
But this comparison has no validation in the argument, submerging a Wool jumper in hot water for 20 minutes (like you would when washing a jumper) is completely different from spraying hot water onto a carpet then immediately extracting it (so it touched the wool for seconds)
You are fond of the cleaning pie, perhaps we should have a 'damage pie' showing what part of the cleaning pie has the potential to damage a carpet. I would guess heat has less potential to damage a carpet than excessive agitation, the wrong chemical or how long we allow the chemical to dwell on the carpet.
Heat correctly used can diminish the reliance for strong chemical or massive amounts of scrubbing.