The solenoid on the bottom opens and closes with 12 volts applied.
When open (12 volts applied) carb GETS FUEL to the carb/main jet.
When it closes (and sometimes gets stuck closed) no fuel makes it thru the carb to the
combustion chamber.
When you turn the ignition on and off you should be able to feel the solenoid click.
If not, pull the solenoid off and what I do, is cut off the pin that moves(or doesn't move).
That'll eliminate that-What it's supposed to do is eliminate back firing when shutting down.
Second, spray starting fluid down the carb throat and see if it lights off momentarily when cranking.
This will rule in or out if it's a fuel issue.
BTW, I have a 2005 Toro zero turn, 42", That I got with a blown up Briggs 3.5 years ago.
After replacing the head and piston, it runs like a top. Absolutely love it. No way would I have a
"regular mower"..
..