"Is that what you
would call a Coil?"
As it says on the label, it is a rectifier/voltage regulator. It rectifies the AC current produced by the permanent magnet under-flywheel alternator and regulates it so the battery is kept properly charged.
Since the engine has mag ignition (NOT powered by battery voltage) you could unplug the unit you've posted a photo of and toss it over your shoulder and the engine should continue to run just as before. But the battery won't be charged and eventually the electric PTO clutch is gonna drop out from low voltage.