The washer in your pictures is between the bearing hoc and drive gear, which is the part both I and IL say you need. You can question what we recommend, as you may have more experience than we do with Toro mowers, but I and I’ll bet IL will stand by our recommendations. Maybe someone smarter...
This manual will tell you how to hook up, test, and adjust the governor. https://www.mymowerparts.com/pdf/Kawasaki-Service-and-Repair-Manuals/FR651V-FS651V-FX651V-FR691V-FS691V-FX691V-FR730V-FS730V-FX730V-KAWASAKI-SERVICE-REPAIR-MANUAL-99924-2093-01.pdf
First, check to see if you may have installed the coil upside down. Second, set the air gap at .010”. Check for spark with kill wire off, if spark attach kill wire and test again. Report back findings, with unit model and serial numbers.
You’ll probably going to have to check under the unit, I’m thinking bad pulley on the drive tensioner arm. Post unit model and serial numbers from under the seat.