Greetings, I read your question and I think heavy-equipmentmanual.com can help you in this. Here's the site link - https://www.heavy-equipmentmanual.com/
limited info generic answer multi reasons --kill switch fail--fuse-wire/connector fail --battery--ign switch --etc ---do an online search and include year quite a few free service manuals for kawaski
this video's may help you with you're problem. \/ Also for pulling the bearing out of the axle I recommend a bearing puller set you can buy cheep online and rent a slide hammer from most auto shop's another video about the tool. \/ and use a socket that has the same outer diameter as the bearing to install.If the rear axle is not the same please disregard thisbut the video may still give you some insight as of whats ahead.
is the ignition switch bad? if it has been working fine off of the 2 wire switch,thats what i would replace with.if it worked with 2 wire switch why would you need 4 wire switch.
I had a similar problem. pull start worked no problem, and starter only sometimes. turns out the starting system as a whole (starter, old wires, solenoid etc) were drawing so much power that it wasn't allowing much for power for the coil and spark. it was too weak
follow it until you find the short. the wire may have rubbed raw against a metal part. Happens a lot. remove the fuse and put an ohm meter to the frame and to either side of the wire where the fuse mounts. It will tell you which direction from that point to follow