You should have drained the tank, changed the oil, cleaned/oiled the foam filter and checked all fasteners, including the head bolts and engine mounts, then fuel it with fresh gas with some Sta-Bil and a little Marvel Mystery Oil in it. Still, you actually got away with it, I think. I suspect the exhaust valve is not closing enough. There could be some crud on it that broke off in the run, or a broken spring. Sometimes the valve stem just sticks from bad fuel. I would definitely look at the flywheel/key situation first, then pick up a head gasket and remove the head. Turn the crank and see if either of the valves doesn't close. I fixed one a long time ago when the valves stuck by putting a fuel system cleaner in the gas and shooting it toward the valves from the spark plug hole. Over a few days I would repeat the process, pulling it through for a few times. I could feel the compression coming back and then it started. I got it warmed up and changed the oil, adding some MMO to the oil. Did I mention how MMO keeps mowers running clean? :)
Since it won't run on gas or starter fluid, I'm thinking it is not a carb problem. Assuming it's a 'Pulsa-Jet' type, the carb is very simple but has one part that goes bad - the diaphragm fuel pump located under the carb flange at the base of the carb where it is bolted to the gas tank. But again, if it were bad the engine would still run on starter fluid or gas.
One more, and a quick check - put a wrench (ratchet/socket) on the head bolts to see if they are loose - this usually causes the head gasket to burn out. Even if they are not loose, the gasket could be leaking. Get somebody to pull it through slowly to see if you can hear a compression leak,

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

Points? Ye gods. I don't even have tools for points anymore. That mower has to be 30 years old!

The engine is 40 years old. Note the Briggs numbers the original poster showed: Code 73050802.

Briggs didn't do to Magnetron until 1983 or 1984.

The mower could be any age, but the engine is 41 years and 8 days.

Amazing! Its really hard to keep one going that long.