Thermal fuse seems like a good description. It it is connected to the circuit powering the main fan. Bypassing this, causes everything to work fine.
The man who wrote this is a house-guest of my friend who is Chinese-American. The house-guest repairs things like this for a living in China but does not know the English words for electronics components. My friend speaks limited Mandarin and does not understand Chinese names for the components.
If it is only connected to the fan, it is a poor design, but if it cuts power to the whole unit, it is in there for protection. It would be a thermal fuse.
Similar devices are used in waffle irons. They look a little different, and have the ability to turn back on once the temperature drops. This does not sound like the intended behavior here.
Thermal fuses are generally used in devices that *may* generate excessive heat if something misbehaves. Once they trip, they are garbage and must be replaced, but it is a good idea to buy a spare and keep a close eye on the device to make sure it operates properly.
That part may have expired due to stress and age, or the device you are repairing could have another problem causing it to overheat.