This may sound counter intuitive, but try replacing the spark plug. those .20 thousands gap plugs are bad for fouling and doing weird things. Not on Briggs, but have seen similar issues on Honda engines that would start and die, or would start and run 5 minutes and die, clean the carb and would run another 5 minutes.
Also something else to keep in mind is those engines are very fuel quality picky. To the point the gas that you purchased yesterday is too old to run today. Sarcasm to a point, but common enough that if one comes into the shop that doesn't start the first thing I do is dump the fuel. The fuel that runs in every other piece of equipment that you own may not even begin to start with the newer Briggs push mower engines.