If the tank is below the inlet to the float bowl, it must have had a factory installed pump. Most B&S had a pulse pump, operated by the pulsation in the crankcase caused by the piston moving up and down. If the poppet valves or the diaphragm fail in one of those, or you get oil behind the diaphragm, it may not work.
Replacing with an electric pump is likely causing too much fuel pressure, overcoming the needle valve and allowing the bowl to flood... and you see the slight hurricane inside the intake that makes the engine run like... you don't want it to.
I would return the electric as it has too much pressure, and replace with a 'generic' pulse pump, or one designed for the Briggs. A pressure regulator may do the job, but you are doing nothing extra except adding complexity. IMO. The simple previously there worked for a decade or so..