According to their documentation, power is fed directly without pressing the button. Pressing the button interrupts power to the ignition circuit, but only while depressed.
So with key ON, you should have 12v to the coil. If not, resolve that first (make certain you ground the coils to the frame via the mounting bracket).
Once 12v to coils is solved, then you deal with “disruption” via the push buttons.
**** On a personal note, I think this is a TERRIBLE design if I understand their documentation clearly. The KILL switch only disrupts power while depressed. So you running along, need to kill the motor in an emergency, press the button, motor stops. You let go of the button, what insures the motor doesn’t restart? Simply that the starter solenoid isn’t excited? Hmmmm…. Makes me nervous to think about the possibilities.
Push button KILL switches as a stand-alone control are not great. A toggle/rocker type would be far more prudent. Flip it and power is disconnected. Flip opposite, power is now available. It’s a more static condition. And how can you rely upon a stranger who may come upon you after an accident, bike running, and they not immediately see how to kill the motor? Just saying… You might want to give that a bit more thought. ******
But your wire schema sounds correct per their diagram, just confirm power as described above.