power only gets to the starter button when 1) the key is switched to [on], 2) the killswitch is set to "run" and 3) the brake lever is held down (prevent you from starting it with throttle accidentally open and it just taking off)
Anyway sounds like you'd just need to bypass the starter button to just go straight from battery to button and back to the starter relay. you can always jumper cable the starter direct from the battery to see if it works, like bypass the relay to check the starter function.