You need a close look at all parts of the wiring. The most likely source of smoke without fire is electrical insulation being vaporized.This leaves the copper wire uninsulated and far more easily susceptible to a short circuit that can do unexpected damage or disable critical circuitry. It is most likely from a high resistance short from wearing through against a part of the structure or a place where the conductors have fatigued and multiple strands have broken, leaving the few remaining strands overloaded. There is also a possible overload elsewhere, perhaps in the engine, that is intermittent, and that overloaded a wire behind the panel.
I agree that checking capacitors for failure is a good idea.
What does the instructor expect to be normally so hot as to vaporize oil? Find a good mechanic who does electrical work.