That is the exact problem I had with my Doodle Bug, in fact, that is the problem that brought me to “OldMiniBikes.com” looking for answers.
Thanks for posting the video; I was not able to describe the problem like your video did.
My Doodle Bug had this problem the day I bought it. My problem was, the chain was too loose and the idle speed was too slow. As the chain vibrated, and swung up and down at idle, the clutch was grabbing thinking it was getting throttle. When the clutch grabbed, there was not enough idle speed to keep the engine running and it would die.
Tighten the chain (1/2” to 3/4 “movement). See if that helps. As other people mentioned, lift the back wheel off the ground and give it some throttle. Once the engine warms up, it is far less likely to happen.