It's mostly a pinch point on time vs prioritization of tasks flying the airplane. If EGT% is pegged out at max, you don't have a chance unless you shut it down as the turbine will melt, not just overheat. Sometimes you don't have time to even care (as heard in CVRs). If the egt is simply overlimits you might be able to recover some thrust by reducing the throttles or restart if you have the time, but otherwise just leave it alone and work higher priorities (damaged engines rarely cause significant hazards after the first seconds)
If you have time to even think about the engines, then consider the restart itself needs time (airspeed, altitude or APU). The only problem I can see in past history has been the prioritization of tasks or not reducing the throttles for simple overheats