You have evidently gotten the timing gear out of time! Easy to do when reassembling after opening the case. I use a piece of masking tape, or heavy thick wheel-bearing grease to hold the cam in place, then carefully lower the sump into position!

Well you do not need to remove the flywheel to remove the starter. Air gap is .010", Be sure the mounting surfaces of the coil AND the engine block are bright clean and the "Kill" wire hasn't been pinched under the cooling shroud. Walt Conner

This is not the first time one of these Twins has misbehaved this way. Because the single magneto coil has just one output circuit that powers BOTH spark plug cables, BOTH cylinders receive a spark at the same time, every time the flywheel magnet passes the magneto. This is characteristic of a flywheel controlled (or flywheel driven) ignition system. It is called "wasted spark" ignition timing because a spark occurs on the power stroke (which IS NOT wasted) and on the exhaust stroke (which IS wasted). The point of all this is: Electricity will follow the path of LEAST RESISTANCE. The coil is powerful enough to make a spark jump at both spark plugs simultaneously when the coil and both spark cables are in good health. When one spark cable deteriorates and becomes more "resistant" to the flow of a current, and/or the coil produces a weak spark current, the entire spark current may flow through one of the cables while the other cable flows nothing. In the early stages of this ignition failure, you may observe exactly what you have been seeing. Erratic spark delivery accompanied by one cylinder wanting to cut out at times. To make matters even more complex, combustion chamber pressures and the air/fuel mix ratio can affect how a spark will, or will not, occur in the cylinder ( see in OP where the right plug fires outside of cylinder but not when installed). Here again, the power of the coil and the resistance of the spark cable are also part of the equation. Using the Briggs spark tester is more conclusive of whether a plug is actually firing when installed in the cylinder than removing the plug for spark check because the Briggs spark tester accounts for the influence of compression on the spark circuit. Open air tests for spark do not account for that influence. If your combustion chamber has much carbon build-up, you need to decarbon both cylinders and replace the head gaskets. If, after doing the decarbon work, you get the same loss of regular spark.............replace the magneto.

I just read this and it sounds to me like it may well be the woodruff key in the fly wheel that sits on the crankshaft has moved a little or even snapped......Some times as you mow and it say hits a bit of wood or comes under heavy load that key can move a little or as i say snap. Then when you try to start it you may get a kick from the starter pulley like a horse kicking you in the arm...I would take the top pulley off and the NUT then look down and see if the KEY is in line with the pulley SLOT. If its slightly off line even 3 or 4 mill of line i would take the flywheel off and renew the key or check it and reseat it....It may be some thing else but if you have tried every thing and it still kicks check that..Its quite common...It just put the timing out a little........................................I have a 6 year old Sears Craftsman lawn tractor with a 24 HP OHV Briggs & Stratton engine. When I was moving the lawn the other day I heard a "pop", then the engine died. Now when I try to start it the engine kicks and backfires, but never comes close to starting. We checked the fuel and the spark and both are present. I can't help but think if it died suddenly, and now is backfiring when I try to start it that it must be something related to the timing. Unfortunately, I don't know anything about how the timing works on that engine. Is it possible the timing chain/belt (if it has either) could have jumped a tooth? Any other suggestions for what this could be, and how we would go about diagnosing?