If you're doing something with the throttle cable, and it starts right up when you do that, then I think that points away from it being an electrical issue.
When you first reported the issue, you said the engine would die after running at full throttle for 5-10 minutes. After replacing the thermostat is it able to run at full throttle for a longer time? Also, does the engine ever die when it's at full throttle, or does it die only when you return to idle?
Also, what are you doing with the throttle exactly to get it to start back up again when it's still hot?