If you have to ether a motor it's likely because the fuel pump diaphram is hard. The side of the motor has a pipe coming out of it, under that plate is a diaphram that acts as a fuel pump, it fills a "float bowl" inside the fuel tank that the carb draws from. If that motor will only start if the tank is full that is usually why. If your fuel tank is remote from the carb then my solution won't work.
I serviced a gillion of the Briggs and Tecumpse motors in a fertilizer business over ten yrs. Get rid of point ignition and keep the tank full solves %80 of woes, keep oil changed and air filter gets another %15, last five seems to involve uh, farmer inputs, such as a wire on the governor or other high performance mods!
If something breaks it was usually cheaper to get a new motor and pump assembly than try and repair. I've found most Briggs will outlast the piece of equipment they are attached to. The better equipment gets a better grade of Briggs, so the rule holds true no matter the line generally.