The difference I see, is that in the OP game every player is guaranteed to give out $1 at each step, while in the video, a player is chosen at random to give out $1. I don't know how this might change the long-term distribution.

Insightful point, especially considering what the difference might mean on the physics side. Aka a time evolved system of uniformly every particle that has energy donating some energy to another random particle vs random particle donating energy to another random particle.