OK good, then the venting doesn't appear to be the issue at this time. So next you need to determine that the carburetor is actually running out of fuel. You stated earlier that you drain the fuel from the carb, so I take it you know exactly where the drain screw is ?
So here is what I would like to know ?
1.) when the engine stops, does it start RIGHT back up, or how long before it will restart ?
2.) so immediately following when it stops, if you can get to the drain screw on the float bowl and open it to see if you get any fuel out of it ? That would tell me in actuality if indeed the float bowl is full of gas or empty ?
Also let me point out that that this unit has a vacuum controlled pet-cock that basically turns the fuel flow on when the engine is running and then shuts off when the engine is stopped, that valve is controlled by engine vacuum, the diaphragms can get old and brittle and then cause problems.
So before throwing parts at it, I think if you can answer the questions I thrown at you that would determine if we need to go in the route of the pet-cock. BTW those are about $18 retail. Also I'd like to mention that I'm not ruling out the possibilities of an issue inside the carb, or an electrical issue in the ignition system ?