The reason I came up with that generalization is because engine temp is definitely not going to affect the gearbox, just like a car it's going to be limited by the amount of torque. Re: the engine, if the compressors and turbines and spools can all support the torque then that wouldn't be a limiting factor. The amount of torsion that these parts could withstand before deforming is immense. The only real limitation I can think of would be the engine temp; if it gets too hot like you said it will just melt and destroy the engine. From what I understand, engine temp limits can be based on TIT, ITT or EGT. What's the reasoning when selecting a main temperature to monitor? I would think TIT would be the most important to monitor. I don't think all airplanes with turbines have all those engine temp sensors.
Re: the gearbox, engine temp really shouldn't be a factor when considering operational limits. Torque seems to be the only limiting factor.
So if you're operating at high altitude in cruise and you have really cold temperatures, your main concern would typically be the torque in my opinion; you'd hit your torque limit before you hit your engine temp limit. But if you're taking off on a hot day you'd most likely hit your engine temp limit first. Any turbine pilots that can comment on that?