I had this exact same issue with my 98 superjet. it would run perfect and start right up when I would first start it. But if I was to ride it for more then 10 mins and shut it off, it would crank over but not fire. I would wait 20 mins and then it would usually fire up again. It was weird, I checked for spark before I went out and did not see spark but it would still fire. I even checked it in a dark garage. I also first though it was a fuel problem and checked everything and put a new one way valve on, still did the same thing. I even went to the point of putting a new stator in, no change. Then I put a different starter and ingetion coil with new plug boots, no change.
Eventually I got fed up and just bought a good condition complete ebox and put that in the ski and it ran perfect and did not have the issue anymore. So the problem was one of the electrical parts I did not replace before switching eboxes. If you can get a clean complete ebox for under $200, I might recommend that. And later on if you have time, test out the electrical parts one by one, to see which one is causing the issue. If I had to guess I would say it was the CDI. Just never had the time or the desire to go back and check the electrical parts like I suggested.