I have a 2005 Mongoose 300 quad which is now acting just like your scooter was in your first posts.
It will crank the starter great, engine fires, runs great for three or four cycles, then dies quickly, just like you turned off the key or pressed the kill switch. No slow sputter like poor fuel supply, but a quick die.. I am in neutral and the parking brake is applied.
I too have done (most) everything mentioned throughout the prior posts, with no good results.
But I have stumbled onto a new possibility - possible failure of my “gear shift control assemblyâ€.
Does your scooter have a similar safety device? My quad is designed not to start unless it’s in neutral. Even with my shifter in neutral, my quad acts like your scooter- starts but quickly dies.
If I do not have my “parking brake†applied, the starter will not turn at all. That’s another safety feature built in that I can live with.
To get it to start, before pushing the start button, I move my shifter from neutral to forward, then back to neutral, then press the starter button, and sometimes it starts and keeps running. If that fails, I shift from neutral to reverse back to neutral then press the start button and sometimes it runs and keeps running. It’s hit and miss.
Not sure this helps, but it’s the least I can do since you guys have been working so hard trying to resolve your situation which has very similar symptoms to my quad.