Without a wiring diagram in front of me I wouldn't know for sure but all those interlocks should be a series circuit, a break anywhere will open the circuit. With a test light or a voltmeter start testing and see where power is or isn't then follow the yellow brick road.
Padroo has it backwards.
The kill circuit is a ground circuit that grounds out the coil, so a break anywhere in it will allow the coil to fire, not prevent it sparking.
There is 2 ignition circuits in play during cranking
1) key switch
2) Parking brake & PTO switches
Depending upon your model, the parking brake switch is a double switch.
One pair of contacts will have 12V pass through it which goes to the starter solenoid & the other pair ( usually black or brown ) will e a ground circuit that grounds out the magneto.
The PTO switch will be the same as the brake if you have a manual blade engagement .
If it has electric blade clutch then the PTO Switch will have a lot of ground contacts + a pair of cranking contact ( 12V in start position ) + a pair of Clutch contacts , usually a ground contact as well ( Electric clutches are electrically isolated so have a + & - connection )