Yes, it is perfectly feasible to grind dungeons to level. The problem of course is that you need an in-demand role to do so; as a healer (my experience) and as a tank (probably), there is zero wait time (usually) for a dungeon. That's better than sitting around in a party waiting for a FATE to pop up near you.

The XP from a dungeon doesn't come from the measly 2k or however much experience you get for completion, it comes from the monster kills. In the time it takes to complete a FATE, you can get 3 or 4 times the experience clearing mobs in a dungeon.

Then of course there's the fact that grinding dungeons gives you green/aethereal gear, which I notice in this game can last a good ten levels or so before it's outclassed by standard gear.