My objection here, is that people seem to mix 3 things together: Engine fire, severe damage and engine separation - just because it is the same same steps that are necessary (hence same checklist) to bring the situation under control.
"Hey man, you got to idle that engine if you got an overheat, otherwise it may lead to fire and then you'll get severe damage and your engine might fall off" - might seem like a logical train of thoughts. And probably could happen: Overheat of gearbox leads to fire, fire ruptures oil supply to the engine, bearings fail, causing severe internal engine damage, high vibrations cause engine mount to fail, engine separates. But for this to happen the flight crew would have to sit on their hands for several minutes (maybe 10s of minutes for all the things to develop), hence a very unlikely scenario.
Fact is you can have a overheat fire, without severe damage. Or you can have severe damage with out a fire. Or the engine might separate with out either.
An engine mount can fail without previous warning.
A fire in the cowling can take place due to a fluid leak.
An overheat condition can take place because of a bleed leak
A severe damage can be caused by a bird.
Etc.
The point is, to THINK about the situation, before reacting. In simulator we are taught drills - predetermined responses to predetermined situations. Which is not bad at all, because it's good to have a set of responses ingrained in your backbone.
But it is still useful, to think, and know what is going on. I.e. an engine (cowling) fire can burn for quite sometime before action is required. This is useful to know if you depart with critical terrain. The side with a cowl fire still has an perfectly good (internally) engine. It might make your life easier, to let it burn for 1-2 mins, and use the thrust to gain altitude, before attending to the cowl fire.
On the NG,the sensors for overheat and Fire are located about the Fan and the engine Core...so quite a good idea to idle the engine..
Is a false statement, if I understand it the way it comes over. I.e. "reducing the thrust causes the core temperature to decrease and hence the overheat condition may disappear". Again, the temperature inside the core, is
not being monitored by the overheat/fire detector loops. It might be a good idea to idle the engine for other reasons (like a bleed leak).
Follow the qrh next time. Is better to shutdown an engine because a false alarm and not to have a severe damage thinking that is a false alarm.
Is a false statement. First, it won't go overheat -> severe damage. Considering the light came on off, it would probably either be an indication fault or the temperature had barely risen to trigger the overheat condition, hence still a way to go before the fire loops would trigger, as they have a higher threshold.
Just because the simulator is programmed to: overheat condition x 15 secs = engine fire. It doesn't mean it will happen like this in the aircraft.
An overheat flicking on/off like in the original posters scenario, might not require an immediate engine shutdown. The poster thought about the situation and monitored it, without acting - in the end the indication disappeared. All good and well.
In short, I think the original poster did a good job. Probably getting the aircraft to it's destination, without making a big drama.
N1 Limit, did you get information for your maintenance afterwards, as to what triggered the overheat condition?