Old thread, but I ran into this today and found a partial solution.

While the pre-5.5 quest history dates are incorrect, you can see the correct dates of the achievements that were associated with certain quests. For example, the final ARR quest grants the achievement "A Warrior of Light". In your character profile on the Lodestone, just go to Achievements > Quests > Main Scenario (or whatever quest category interests you) to see the date those quests were completed/achievements were granted. I used this list to get the names of the achievements for each main scenario quest.