I think there's technical complications with founding additional city-states after game start. Things like Diplo votes, and maybe other stuff?
If the question is just "Do you want to see more City-States owned by Barbarians?", then my answer is "Yes", I think it's a fun point of variety. Making the Horde quest harder for CS to self-clear perhaps, and letting Barbarian cities grow more naturally (instead of being stunted severely) would go some part of the way towards this. Without the latter, I don't think it ends up being a good idea for more Barbarians to own cities.
I floated the idea of some percent of Hostile City-States spawning in as Barbarian-controlled awhile back, but the idea isn't fully fleshed out. What I like about the idea is that you have this conquest-related "quest" by default, where you need to defeat the barbarian city to "liberate" it back to a city-state (or just add it to your empire). It's an outlet for otherwise-warmongery Civs picking a target that doesn't upset everyone around them (including other CS).