carb is a suction-feed. There's a long stem that goes to the bottom of the fuel tank. This has a fine-mesh metal screen for a filter, which is clean. There's another stem, shorter with what in a Holley automotive carb would be called a 'jet' staked into the end - this goes into a shallow pan tack-welded to the inside of the gas tank. There's no moving parts in any of this except for the throttle and choke plates. I can't figure out what the purpose of this short stem/shallow pan is or how fuel gets into the pan to cover the jet, but it'd make sense that once the shallow pan ran dry the engine would quit. CY: Briggs had two carbs, well, more than two. I think what you're describing is a Vacu-Jet. When the piston goes towards the spark plug, it creates a slight vacuum in the crankcase. This slight vacuum goes to the carb in a silver tube about 3/8 diameter. Which moves a diaphragm, and pumps a bit of gas into the small pan. The air flow through the carb body creates a slight vacuum (venturi effect) which lifts the gas into the combustion air. On the other side of the motor is a cover over the bottom of the valves. The cover is abotu 1 inch wide, and 3 inches tall. You'll need a 5/16 nut driver, it is held on with two screws. This cover has a gasket, and contains a check valve. When you get the cover off, you'll see a hole in the inside surface of the cover. It is possible this cover is dirty, and not allowing the crankcase vacuum. Gently remove the gasket, and clean out the valve cover with brake cleaner, soak in gasoline, etc.
I don't think the engine runs long enough to get hot enough to cause the ignition system to fail. Where it the system located? CY: Under the motor cover. Next to the flywheel. If it's an older motor, it has a points and condensor setup under the flywheel.
I don't think it's the RPM governor but could check it - where is it located and how can I check it? CY: On top of the carb, there is a spring. And a short rod. the rod goes to what looks like the rudder of a small plane. This rudder (can't remember he name of it) is under the motor cover, and is blown by the airflow coming off the flywheel). These parts should move gently and easily when you try to move them.
Grrrrrrr. I'll make a call to the local small-engine repair shop, maybe they can give me a straight answer. CY: Please keep us posted.
Steve