There is an engine stop (fuel cut off) solenoid and relay that controls the solenoid. The solenoid is on the front of the injector pump. You could check that you have power on the solenoid when the ignition is on.
So the seat switch only controls the solenoid and not the starter circuit? Seems like strange logic, what's the point of allowing you to drain the battery trying to start an engine with no fuel delivery?