Lack of fates to run sounds more like a server population issue, granted on Balmung there's always tons of them around and plenty people doing them. That, and the actual quests in the game do give a fair amount of EXP if you haven't done them all -- I saved the side quests for leveling up other classes, since the main story gave so much experience as it stood.

On tank and healing classes, I just run dungeons. Got a DRK to 41 within a day or two of that and I wasn't really grinding it, just running a few here and there.

I've been with XIV since Beta and don't have anything at 60, though I took a long break here and there. And yet, with leveling other things, I hardly touch leve quests unless I really want to hit the next level for whatever reason.