“Riddle me this, mechanical guys: can a carb’d motorcycle engine run without a battery or regulator/rectifier power once it’s started?”
The answer is in fact NO. Even though in FI engines you need power to pulse the injectors open, Carb’d engines work on vacuum and gravity. The power source that you do need is for the spark running to the coil and plugs. If the battery is dead or has a dead cell in it then yes you could be losing power via that route. Though the only thing that throws me there is being able to constant crank on the start up re tries.
It’s possible you could have an exposed wire somewhere creating a short with the sudden movement forward on stopping. Maybe causing an unwanted ground. Once the movement or inertia forward is gone the culprit has time to settle back in a neutral position allowing it to become a regular circuit again.