Hi Taz, it sounds like you have proved that the fan can work, just not when expected. Plus the AC runs up - did you feel it pushing through cold air? If yes, then the AC is also okay.

The question mark would therefore be surrounding the part with the crumbling coating, which is likely to be some form of Thermistor or switch - under the green coating is likely to be a coil, like in a transformer or relay. If it is crumbling and is located in a heat area, then the coating has succumbed to its temperature surroundings and at worst needs to be replaced.

Jim M