I doubt there is any fuel in the cylinder - it will have long evaporated. Unless, of course, you are pumping it in using the primer (its fairly easy to overcome the float valve in most carbs)? My money would be on the starting system - easy to check if you put a volt meter on the motor main connector with the starter engaged. Could be undersized cables or a contaminated starter commutator/contacter. Turning over a cold engine when its been sitting for a while is particularly difficult for the starting system. The Briggs engines are low compression, long stroke so should be easy to turn over, even for a puny starter,