Has anyone changed out the ignition switch and installed an identical-looking one that is INTERNALLY different and feeds power TO the EI, instead of only grounding it to stop the engine?
With the "KILL" wire disconnected at the engine, there should be spark when the engine is cranked, with NO connection to the electrical system or battery at all.
Below is a link to the official Kohler manual (in .pdf format) that can be viewed or printed for FREE.
There are several pages on how to troubleshoot the EI.
If it IS "smoked", if the engine has the block drilled (and plugged) and screw holes tapped at the breaker point location, a correct 12 Volt coil and breaker points can be installed for a LOT less than replacing EI parts. (NOT sure if ALL engines so-equipped have the required breaker point cam on the camshaft???)
Haven't looked lately, there used to be some changeover kits on Fleabay.