i found / stole this from the net, hopefully it helps.

The wire which comes off the + battery to the solenoid is the big thick supply cable, the other big thick cable from the solenoid goes to the starter motor. there is no flow of power to the starter from the battery until the start button is pushed, this causes the solenoid to energise which causes a copper plated disc to be pulled up to the terminals which the big thick cables are attached to and make a circuit which allows power to flow to the starter from the battery, the solenoid also pulls the mechanism which actuates the starter gear and pulls it out onto the ring gear.

 What a lot of shovel riders do is remove the wire from the relay to the solenoid and run a wire from the + battery to a 35amp tractor starter button and then to the solenoid, this bypasses all the crap and you do not have to worry about voltage drop, every little bit helps starting a shovel. You could try this by just running the wire and touching the battery terminal to engage the starter and if it starts remove the wire and hopefully the starter disengages, do this with the ignition on, what you are doing is bypassing the starter button which i believe is on the same handlebar control as your kill switch, this could be why your bike was cutting out, if you had a short in the switch it could allow the power flow to remain when you took your finger off the start button and cause the kill switch to stop the bike.

Long winded i know but it looks like you have tried a lot of things...............Good luck...........i`m sure someone will point out the mistakes thanks in advance, i do not claim to be an expert.