I think you are being a bit too picky... The T6NL is a "caption" extermely well known to any Hawk pilot (must be if I can remeber it all after all thes years!), and all "bitching betty" does is restate the (prioritised) illuminated captions. Since the pilots would "know" well what T6NL means, seems fine. "Engine Overheat" is misleading, since there are other "overheat" type cautions often present, and why confuse with the "panel" saying one thing, and bitching betty another? In addition, any Hawk pilot will have heard and practised the consequences of T6NL every Sim trip... I certainly did (albeit pre the voice call out).
I was on the first Hawk course (11 Cse) at Valley in 1977. The T6NL warning was supposed to indicate that the temp- or NL-limiting system had failed so that you could take corrective action. It was simpler than the myriad warnings that might have been produced if the engineers had had their way ;-)
I, too, am impressed by the calm way the IP handled this emergency. I shall show it to my students on a ground training day.