FFXIV doesn't have the necessary mechanics to support a Dueling system because:
1) Tank/DPS/Healer trinity have different strengths.
2) Damage potential of certain classes and timing (how casters can be interrupted, but bards can spam attacks instantly)
3) It would be inherently unfair weighting against DPS, since either the tank would be able to soak up more damage, or the healer would just heal themselves and it becomes a race of who can outlast the other.

Let's not even think about support mechanics like the chocobo or arcanist pets.

Other games that have such systems, all their classes are effectively DPS classes to begin with, and fighting a healer or tank is not typically a winnable scenario without beating their cast speed. Full open-world PvP games are designed around "everyone is an equal DPS" because players have to defend themselves.

That said, games that have both PvP and PvE are only good at one of them.