I know in my old kohler (big engine) there is a check valve that can get some debris in it and it will smoke. See if there is one in your briggs.
Around here, I pick up mowers off the curb and they are OLD - rarely is one a smoker (usually a gunked up carb). Briggs or Techumsehs seem to last forever.
Actually, I don't have an engine newer than around 20 yrs. 7 small tractors -3 pushers at the moment.