Thank you BertsMobile1, looks like a bad solenoid. I tested as you described and it started right up.
If the solenoid worked when you jumped the trigger wires at the base then the solenoid is good and the wiring is bad
As you can see from the diagram Star posted the black wire is always ground ( check it ) and the white wire should go battery voltage with the key in the start position , the blades turned off & the brake set .
If no voltage then pull the plug from the key switch & jump from the B terminal ( red wire ) to the S terminal ( white wire )
If the key switch is bad & others are good the engine should crank
No cranking then with the key plug still jumped pull the plug on either the PTO or Brake switch and do the same
The one you jump that cranks the engine is the bad switch.
Note because there is no power to the carb solenoid the engine should not start , or may fire a couple of times & stop
Be careful not to jump the White to gray on the brake switch ( bad choice of colours ) or you will send battery voltage to the magneto & fry the timing chip
You can pull the blower housing off and remove the kill wire from the coils to prevent expensive accidents, I do because it costs me money .
I make up proper patch leads to do these tests which is just a short length of wire ≈ 4" to 6" with an insulated male terminal on both ends .
If all that fails to crank the engine then one of the wires between the switches is broken
Same tests but use a much longer wire to bypass the loom
Note the wiring diagram above is WRONG
brake pedal up has the bottom pair of wires Black & Gray connected while the White pair are not connected
Pedal depressed reverses this , white connected , black & grey disconnected