I was thinking of a somewhat realistic silver coin (perhaps similar to Ptolemaic emblem, but silver instead of gold).

I think the difference would be the direction that the unit is pushed to: With the current pushing, it would move the same direction as the pusher, which means they will probably collide again. With the "clearing the path" mechanism, the second unit would make two steps into the direction it faces/ or into a direction with free space/ or a certain angle of the pushers movement direction, so hopefully avoiding another collision. I'm especially thinking of the elephant sliding another elephant scenario from the videos above. I don't know if this makes sense or if it's too costly but maybe one active instead of passive move could help solving the bottlenecks somehow Yeah, I'm a big fan

As long as you give proper credits to Wildfire Games and respect CC BY SA 3.0 you're in the clear

Reducing hitboxes might solve the issue with the wider path from your first video, but the narrow one would remain. It's like chasing the issue. I think hitbox shape might be more determinant: in tight spaces, vertices complicate rotating and straight sides give too much front for static equilibrium. Maybe elliptical shapes with increased on-center repulsion range to avoid compact tiling might be worth a try (since that seems to be the only problem about them, for now).

What color do you imagine it to be? I can imagine the Celtic's color, I don't know why, apart from the obvious.  The Greek letter in one color(yellow or Orange)  and the rest in other colors including green.