... do you think the premature breakdown could be caused simply by too much heat build-up? ...
Yeah, not even the slightest chance. A simple 2-stage gearbox is going to be very efficient, perhaps 96% roughly. I've measured the temperature of the housing after driving 40km and found it to be about 20 to 25°C over ambient, not even close to the 100°C over ambient it would take for me to be concerned, such as a conventional automatic would have. Another owner confirmed this approximate level of warming a few months back on Reddit.
To give that opinion the 'sniff' test, if you consider the average driving power to the wheels to be about 12kW, you could say losses are (1-0.96) x 12,000 = 480 watts. Think of five 100 watt light bulbs inside an aluminium enclosure of that size with ambient temp air flowing past and you get the picture, it will get warm to the touch but you won't burn your fingers.
Now, there's another scale of view to consider and that's the microscopic oil temperature as it passes through meshing gear teeth. It will be a
lot higher because "working" the oil by squishing it under extreme pressure does heat it up considerably. But it gets cooled again quickly when returned to mix with the bulk oil.
Gearbox oil temperature is based on average temperature so that situation is already taken into account. Generally oil needs to reach 130°C (265°F) in
bulk terms before there's any concern. It also pays to recognise that that simple gearboxes are extremely common in industry and are
very reliable. There simply isn't an issue with natural convective cooling off the housing without any additional support. There are no (ICE car) manual gearboxes that require an oil cooler.
EE, the E-GMP design is focussed around cooling the motor with oil, a bulk supply conveniently located nearby. The GRU benefits from this but adds little to the total heat content. It also removes the need to seal the gearbox from the motor, reducing lip seal friction. Toyota was exploiting these ideas two decades ago with the gen-4 Prius.