No Blue the relay has nothing to do with the kick start. The kick start is a mechanical starting system, the elec start is an electrical starting system. They are not connected in any way, well apart from they both crank the engine over, but they are totally seperate systems and should have no influence on each other.
It has to be a problem in your electrics, either in the wiring, the relay, the startet button, or the starter itself. All the relay is is a heavy duty switch that can take a larger current than the small button on the handlebars, you press the button on the bars and that activates the relay coil which pulls in the contacts to make the circuit for the starter ( or at least thats how any relays i worked with work).
You seem to have ruled out the starter itself being faulty since you said it works on another scoot, so it must be wiring, relay or elec start switch imo.
Can you borrow a relay from anyone else to try? Did you replace the wire that was burning up? Where did that wire come from and go to?
