One article seemed to indicate that you could flip cities with religion.
This makes more sense imo. I always thought culture flipping was a bit odd. There is no way that if we build enough radio transmitters and Opera houses in Dover, Calais will decide to become British without the French government having any say in the matter. A religious revolt against the state (in classical or medieval especially) makes a bit more sense.
Maybe there's some way to war with culture instead of with military force, basically take cities somehow. I don't know how to make that mechanic work nicely though.
I remember that one article said that you can lose city to another civ because of unhappiness...
City flipping with diplomats/spies used to be my favorite thing in early Civ games, but I agree it should be much harder to do.
As for purpose: this will still cause that civ to not like you very much, but might get you a city without suffering the warmonger penalty for taking the city (you will still get penalty for espionage, grabbing territory and aggressive expansion if such penalties exist and the UI will have a solid casus belli against you). Not as strong as in early civ games but if you really want that one city without the cost of conquest and have the extra spy - perhaps you can.
- conquering a city which has the same religion as your civ OR is dominated by your culture should both be much easier,
- civs should offer you cities which feel closer to you for religious or cultural reasons and are therefore unhappy and unruly for a relatively cheap price (maybe even for free if they are a nuisance), BUT
- outright sudden flipping without active involvement of the civ they want to flip to should not be a thing.
I don't think there is a strong reason (except for the fun when it works) without going back to civ2 governments, which is not likely. It can encourage a player to avoid crappy cities that are easily flipped, but I don't think the developers want that.
However if you want such a feature I'd make it part of espionage. If you don't - well, much simpler not to include it. I'd be happy if it exists (because of the fun factor) but would not be to sorry if it doesn't.