Didn't know the Engine Failure or Shutdown NNC on the 737 has 2 parts/sides. I thought it was the same checklist for 2 conditions: an engine failure or an engine shutdown - but the steps are the same of both conditions.
And why would the engine be automatically considered as failed, especially if the OVERHEAT light extinguishes. For example, you get the OVERHEAT light on takeoff, then do memory items, pull the throttle back a bit to let's say climb thrust and the light then extinguishes. Would you consider an engine which is working normally at climb thrust as failed? Just because the engine won't give you full rated thrust it without problems, it doesn't mean it's failed. And FYI, if the OVERHEAT light remains illuminated, you do the Engine Fire or Engine Severe Damage or Separation NNC, not the Failure or Shutdown one.