We start by working out how it is shutting down
I rather think it is electrical from what you are saying.
So remove the blower housing and remove the kill wires from the coil
Remove the solenoid from the carburettor and verify that it works, the plunger should retract when ever there is 9 V or better applied to it .
If it works, refit the solenoid , tie the kill wire out of the way so they wont foul on the rotating flywheel , replace the blower housing & go mow
Problem goes away then you have a safety switch problem with the wiring .
Problem remains then we have to check the fuel solenoid
I use a 9V transistor battery that I plug into the solenoid and go mow .
When I don't have that with me I hook the voltmeter to the battery and watch for voltage drop just before the engine cuts out
From what you have written I would suspect that is the reason for the shut down , finding the cause is another matter .
I used to have some short flanged head bolts & O rings that I used to replace the solenoid but I have managed to loose them thus the alternative power supply.