Your layman’s answer still doesn’t quite do it for me - if you had two masses in space, joined together by a wire, and you span them around their combined centre of mass, angular momentum would remain conserved even if we wound the wire in or payed it out. If I cut the wire, though, wouldn’t the masses immediately shoot off (roughly) in whatever direction they were travelling at the time? That is my interpretation of the “everything seems linear” perspective colordrops offered - it seems like you only get the apparent rotation because of all the other forces holding things together.
edit: I accept that the non-layman’s answer may hold, I just find the intuition a bit off.
edit: I accept that the non-layman’s answer may hold, I just find the intuition a bit off.