Check for any fuel hoses routed too close to a heat source on the way to the fuel pump and/or carburetor.
Have you checked under the sheet metal ducting to make sure mice haven't begun packing stuff in the cavities there which they like so well.
A little blockage of the cooling ducts in summer can lead to a fuel vapor lock situation that can affect the ability of a fuel pump to move fuel.
Also, you need to determine if the engine actually has spark when it shuts off.
Might also want to prime the carb with a small spoon of gasoline when it shuts down...........if you find that the engine still makes spark.
You really do need a bona fide Briggs spark tester to determine whether the spark is actually disappearing FROM THE COMBUSTION CHAMBER. Ignition can sometimes show a spark if you are simply testing outside the combustion chamber with a spare spark plug.
To really know whether a spark is happening INSIDE the CC, you need a real ignition spark tester, like Briggs' tool # 19368.
Attaching the ground lead of the tester to the engine at a point away from the spark plug will show about the same results as laying a spare spark plug on the engine and watching for spark.
But.........after you test that way, next, connect the tester ground lead TO THE SPARK PLUG STILL IN THE ENGINE.
If you see spark indicated on the tester under these conditions it is safe to say your ignition is working right.
You can mimic the Briggs tester by inserting an insulated screw driver into the spark plug boot and then hold the metal screwdriver blade close to the spark plug in the engine.
If a spark still wants to jump to the spark plug tip, the ignition is probably good.

I am unfamilar with this engine, but if it has a condenser coil you will see the problems you have if it has failed. It took several hundred dollars and several trips to the dealer to find that mine had failed.

If you feel the rest checks out- it may be time for a new fuel pump.

I agree with the poster that said the ignition coil would be a prime suspect. This happened with our Murray with the Briggs Twin 2 engine.

Which poster said an IGNITION COIL would be a prime suspect?

I believe knuttle was referring to the ignition coil, or Armature-Magneto as its called by the mfr, when they said "condenser coil." Our Twin 2 experienced the same symptoms and that was the cause a couple yrs ago. Try checking for a spark right at the time it cuts off next time to be certain.

And I believe knuttle was referring to a condenser, which is part of a battery and coil type of ignition system which uses old school breaker points.
As a Briggs engine (31P977), this one does not have ignition points nor a battery and coil ignition system.
Nor have I ever heard of a condenser coil (except as a component in an air conditioner system).

Follow up...problem solved.

Just thought I would post the results after my neighbor finally got it running....

He called out Sears Service to fix it....took 2 weeks before they even came....told him, like some of your suggestions, that it needed a new magneto....ordered it and took 2 weeks to get the part...another week later the service man came and replaced the magneto...charge him $279 !!!!....however the problem was still there!....quits when hot...took another month to get Sears back out...then another week to replace the carb...and finally problem solved!...Sears said that they will refund the price of the magneto....

I will NEVER call Sears for any out of warranty service.....