I agree with you, and most of what the previous poster has said. You probably have one of two problems or both. 1) Your engine has a "Pulsa Jet" carburetor, which uses a diaphragm pump to lift fuel to a constant level well, and from that point carburetor acts as a suction type. In addition to bad diaphragm, either of the plastic fuel pick up tubes could be clogged, plus the main jet and passages behind the jet could also be clogged. I'd start by disassembling carb, soaking or replacing fuel pipes and jets, and definitely replacing diaphragm. Upon assembly, initial main jet setting is 1 1/2 turns. 2) Ignition is the next thing I'd look at. Your engine is old enough to have breaker points and condenser under the flywheel. Remove starter clutch, being careful not to damage flywheel fins or the clutch body. Check to see whether flywheel key is partially or completely sheared. While flywheel is off, remove cover and check points. As you rotate crankshaft, points should open and close, with maximum opening being .020. Make sure points are bright and clean, using an automotive point file, and adjust loosening condenser clamp and retighten. Rotate several more times to check point gap. I'd strongly recommend you consider replacing the existing coil with either a Briggs Magnetron coil (398593 $40 OEM or $25 aftermarket), or purchasing the conversion module (394970 $15 - $20). Upgrading to Magnetron is quicker and easier as you don't fool around with the points, and the new coils are almost bullet proof. Suggest you go to Briggs site and pull down an illustrated parts list for your engine. Enter your engine numbers as 092988-0924-99. Briggs wants six digits in the first sequence, and won't resolve "01" in the third. Use "99" as a dummy. Here is a link that might be useful: Briggs Manuals

Lots of ideas about what is wrong, but the answer lies in testing the system correctly. Behind the LF strut there is a gray connector with a red wire that does not appear to connect to anything. Take a test light and connect it to the battery positive terminal, and to that connector. The relay when turned off actually closes that connection and that feeds the fuel pump and is called a fuel pump priming connector. If the test light does not turn on the circuit is open or has excessive resistance. When you turn the key on without cranking the engine the relay will be turned on for about two seconds and then turn back off. The testlight should turn off, for that two seconds and then turn back on. Tell us what happened with this test, and then I'll tell you what to do next.

here is another way of describing what the others have stated. pushing the primer bulb pumps air into the float bowl. A small amount of fuel is displaced through the red tube into the venture area. This provides fuel for starting. When the engine fires, fuel continues to be drawn into the airstream in the venture. The main jet meters the correct amount of fuel and the engine is happy. In your situation, the primer is not delivering the correct amount of fuel. A puff of air provided airflow through the venture and drew fuel from the floatbowl. When emission requirements were developed, Tecumseh designed a primer that was right on the edge. If the fuel level in the float bowl was not correct, or if the bowl gasket had a tiny leak, the primer will not deliver enough fuel. Additionally, the primer bulb may also have dried up and collapsed slightly. O rings on the venture tube may have dried up and cracked. Ethanol fuel contributes to deterioration of the rubber parts.

Oh, good idea! I just did a visual of the oil fill tube without taking it off. I cannot get fill tube off until I get the engine cover off after I retrofit/cut the metal frame...plan on doing this retrofit this coming weekend. In any case for now, it does not appear there is any cracks from what I could see/feel nor did it look like there were any oil leaks in the fill tube area. See attached pic. Since I do exactly know how it works....Are you saying that if there is a crack, this would cause a loss in crankcase vacuum, and this lost of vacuum could in turn cause oil to come out around the fuel pump? Thanks for the idea and I will get to do a full inspection of the fill tube this weekend.