Since we don't know how this thing is wired, we are just "talkin."
But get your test lamp. Should have power on both sides of the starter button with button pushed
Should have power on one or other of the "push on" terminals of the starter relay.
IDENTIFY which "push on" terminal of the starter relay is getting power, if it is, with the button depressed, and GROUND the other terminal. It should crank. If not, bad starter relay. If it does, problems in the NSS circuit.