Of course you don't have to level a tank to complete your anima. Although as a tank you can run the dungeons according to your mood... Slowly, massive pulls etc. and no one can force you to do otherwise... However, many tanks fail because of their confidence in tanking and when they pull 10 mobs they die and then they blame the healer (you). Well for the leaving part, i can understand that some dungeons are really boring to tank for example. aurum vale or tam tara but its just stupid if they wait for 30 min to requeue when they can complete every dungeon in ~20 min.
Since the anima requires dungeons i suggest you find a tank from fc or friend and u will have 0 problems.