This could be a long shot but it happened to a 98 500 Sortsman that was given to me. First off this machine I got needed major repairs. Went thru carb, new cam, rocker arms, new batt, filter plug, oil change, etc plus a bunch of non motor related repairs. Anyway the first time I started it it fired up fine(electric start) I ran it outside that day and that night it would not fire, but would fire with the first pull on the recoil. I started looking at it and found the starter only had one bolt holding it on and it wasn't tight. I replaced the bolts to it with the correct ones and it has never had a problem since. Ever.

Years ago I also saw a couple 500's that did this same thing. I found the magnets in the flywheel were broken but didn't destroy the stator. Same prob.

 

Found the problem.
The pulse coil had been replaced by the previous owner. The pulse coil adjustment should be between .016" and .040". When I checked the gap early on, was close to .040". I attempted to adjust it, but every time I tightened the retainer clamp, it moved right back to .040" so I left it. Was told by an old timer that when pull starting, force on the crankshaft if from the 9 o'clock position moving upward. When cranking with the starter, force on the crankshaft if from the 12 o'clock position moving forward. There is a slight amount of play in the crank bearings. Not enough to feel or hear, but there is a couple of thousandths there. Removed the pulse coil, made the holes in the bottom mounting plate a little larger, adjusted to .020". Starts like new!!
Appreciate the ideas.