Sorry for the confusion.
Coil = magneto, the thing on the other end of the spark plug wire,
The whole system works by a little chip hidden inside the coil grounding out at the correct time to make a spark from the magnetism from the magnet.
IF the black wire is grounded the whole thing shuts down & the mower stops.
Down the bottom of your diagram is the connection table for the key switch.
you will see OFF = M + G.
So if grounding the M wire stops the engine and your engine is stopping then something is connecting that black wire to ground when it should not be doing it.
So you now need to find the rogue grounding point.
It could be a bare wire it could be a bad or incorrect switch.
Because you can start the engine we know that the ignition ( key ) switch is working properly, although you have not said that you can turn the engine off by turning the key off.
So from there we are looking at the other things that connect to the Black wire on the M terminal of the switch to ground
That black wire has 3 connections ( 4 if you include the key switch )
From the key switch going down the page
1) connects to the A terminal on the PTO switch
2) Connects to the brake switch
Both of these switches also connects to the seat switch relay and connect to ground via the 87A terminal connecting to the 30 terminal if no one is in the seat .
So when the PTO is turned off and the brake is down there should be nothing grounding the magneto wire , the black wire that goes to the M terminal on the key switch
We know this is correct & working properly because the engine starts.
Note you can start the engine with no one in the seat, this is so fools like me can work on a running engine.
IF you turn the PTO on , the PTO switch connects A to D which then connects M to Ground via the relay terminals 87A & 30 stopping the engine
If you release the brake the same thing happens, the unnamed terminals on the brake switch complete the circuit to the relay 87A & 30 to kill the engine
When the engine is running you should get voltage down the orange A2 wire to the seat switch then through the switch to the relay moving the contact from 87A to 87.
So now there is nothing connecting the PTO kill circuit and the brake kill circuit to ground so they can not stop the engine.
Now I hope you can follow what actually happens, they do things like this to prevent idiot morons striking a blow for freedom by unpluging the seat safety switch.
SO having gone through this for you we can do things a little differently
Flip the seat and start the engine.
Unplug the seat switch and check for 12 V at the plug, only one side should show voltage, engine running.
Now plug it back in and either sit on the seat or depress the switch with your hand and check for voltage at the other side of seat switch plug.
If you get voltage both sides with the seat switch activated, unplug it and jumper the plug.
Now go to the relay ,unplug it and check for voltage on the green wire ( terminal 85 ) .
Switch the test lamp to the + terminal on the battery and check the relay plug wires for ground, you should only have ground on the 30 terminal.
Now replace the relay and check the 87 and 87A terminals for ground by poking the probe in the back ( wire side ) of the plug.
Should have ground on 87 and no ground on 87A.
Pull the seat switch jumper, the ground should switch from 87 to 87A
To avoid confusion we will leave it here for now, do the above and get back with the results.'
Where we go from there will depend upon what you find.,