I never knew that barbarians take money instead of capturing a city.
1. That is a great idea, barbarians rarely remained culturally relevant, and usually just plundered the landscape (think of the Vandals or Tartars).
2. It is hilariously dumb that I never encountered this in all my time playing Civ V. Barbarians are too wwweeeaaaakkkk.
To expand on the idea of a great chieftain, it would be cool if they created an almost new, "quasi" city-state that was unit and chieftain based, instead of city based. It would have a leader that would come to the edge of your civ and demand tribute before moving on. You could declare war on them jointly with other civs in the area when their tributes become too demanding, you can ally with them, etc. Obviously, by nature, they would be untrustworthy barbarians, even more shady than the normal AI so be wary in dealing with them. Who knows, maybe if they capture a city the could become their own full-fledged city state, representing new political leadership over the same cultural region (like Mongols over China), resetting all favor.