Faulty switches shut down the spark & the engine dies like you have turned it off, cause the switch has actually done that.
Coughing & spluttering will be an engine problem not a mower problem.
If there is a bit of white smoke from the exhaust then most likely a leaking head gasket.
Start the mower, set the brake then try moving the governor arm by hand.
If you can accelerate smoothly then the engine is fine
If it does the same things as when you turn the PTO on then as above, suspect the gasket.
If the engine accelerates fine then that leaves the electrical load.
Check the alternator output.
A low output will cause the voltage to drop which can cause the carb solenoid to flutter
