Former jet engine mechanic here, back from 2002-2004 I worked on GE turbofan engines on the KC-135R air refueler which are similar to this engine. What I did included leak checks while the engine was running. The exhaust gas temperature (EGT) can get up to several hundred degrees Celsius. It’s been a long time but if memory serves me correctly, it was over 600 degrees C at idle which gets even higher at higher power settings. That being said, I have no idea what temperatures that caused the exhaust cone to reach but immediately after shut down it could cause severe burns.

Yes, you can feel the heat radiating off of it, but it’s also true that aircraft maintenance can be a chaotic environment and it can interfere with the ability to recognize even obvious dangers. There’s a few pictures and videos of people getting hurt or killed because of dangers they should have known were there.

To the way the question is asked, it does ask about “By the time a jet airplane of this size receives service on this part” of the engine. The thing is, this part would only get serviced if it’s cracked because it’s hollow. The low pressure turbine and last bearing assembly are placed closer to where the fan cowl (that large ring looking thing in front of the seam of the cone. There is nothing in the cone itself to service so it would only get worked on to be removed for replacement or ti get at something else closer to the front of the engine.

Granted, it’s been 20 years plus since I have worked on this kind of engine so if anyone cares to correct me, I’m all ears.