If your start button wiring is shorted you are hitting a kill button. Disconnect the starter button from the harness and jump across the solenoid with a screwdriver and if the bike starts and then dies after removing the screwdriver then it's not the starter button, but i'm betting it is.

My only remaining guess is the starter circuit relay. If that isn't it, I'm beat. After you mentioned that disconnecting the kill switch nothing worked, I went back to the wiring diagram again. It doesn't show the configuration in the run state, so I can't guess what is open and closed. Obviously from what you said , the kill switch must be closed rather than open as I thought. I see the starter switch also connects to the igniter ( CDI box). If the starter circuit relay isn't the culprit, that would only leave the CDI box , and that is an expensive box to replace. It looks like it may be time to bite the bullet and see a Kawasaki mechanic . I've exhausted my guesses and without being able to physically get to it with a meter and service manual , I'm beat.

Check the engine stop switch to see if it is stuck on, and also trace the wires. We may have misled you on the relay.. There are two of them. One is the starter circuit relay and the other is the starter relay itself.. If it is the relay , it will be the starter circuit relay. If that is the one you replaced, check the engine stop switch. According to the schematic it should be open when the engine is running. Closing it opens the ignition circuit and kills the engine. Holding in the start button feeds power to the ignition circuit until you release the button.