engines are supposed to be hot, the whole engine and pylon is designed to withstand heat, and the flames (if there are any) are being blown clear of the fuselage and tailplane
Interesting personal observation although I very much doubt if this advice will be found in any manufacturer's manual if only because of litigation reasons.
Personally, any fire in an aeroplane that I am flying either as a pilot in command or as a passenger, would scare me stiff. I recall some aircraft experiencing an engine fire have about two minutes before the main spar burns through due to the blow-torch nature of the flame-front and then it is curtains for everyone.
It begs the question why deliberately let an engine burn following an engine fire simply to hang on to more thrust when if the runway analyses is valid the aircraft has more than adequate performance to meet certification figures.
You may never know how bad a fire is until it is too late to fix. Your company policy of delaying action until third segment height is attained sounds like the work of a new broom check pilot sweeping clean without regard to airmanship. Which reminds me. I understand the term "Airmanship" has been discarded in new publications in favour of the rather grand and verbose term "Threat and Error Management."
Airmanship will still do me, thank you very much.