Is there anything about Greek epics and "Gilgamesh" that audiences in those historical contexts would find appealing or reassuring?
Almost nobody (to my knowledge, anyway) writes epics any more. A question, then--why would this form of narrative hold such appeal to the ancient Greeks and the first audience of Gilgamesh? Is there anything about these works that audiences in those historical contexts would find appealing or reassuring?
Homer's Aeniad, his epic about a Trojan prince who survived the Trojan War and goes on to found the city of Rome, begins with
I sing of arms and of a man
Perhaps the most memorable beginning verses of all the Homeric epics, it epitomizes the principal themes in epics--war, or its after effects, and the struggles of a man. The use of "sing" is meant to connote "celebrate."
Because life in the Bronze Age, the general time period for Homeric epics, was, for most of the Greek world, one short or long war after another, war and all of its elements--bravery, cowardice, honor, glory, strategy, fighting skills or the lack thereof--were compelling subjects for the society as a whole. It is safe to assume that no one in the larger area Greece and North Africa escaped the effects of warfare, and civilians were as negatively affected as the combatants. Their interest in every aspect of warfare--and particularly the heroes upon whom they relied for safety--was intense because war was ever present in their lives. Gilgamesh, too, reflected not the struggles of of war, but the struggles of man with gods and finding his rightful place in the world, a problem that, like warfare for the Greeks, was a constant concern in the Mesopotamian world.
What makes the epics so central to the culture of the time is partly a result of the oral tradition. Very few cultrally-important stories were in written form during the Bronze Age, and few people could read, so what we now call "literature," the story of a culture and a race, was presented orally from the poet's memory. And because everyday, mundane life, was not a fit subject for memorizing and disseminating, poets like Homer concentrated their efforts on culturally-signifcant matters, in Homer's case, epic actions and mythology.
Aside from being reassured that there were men willing to do "violence on their behalf" (epic heroes and combatants), the audience was being inculcated in those attributes that the society most treasures--bravery, honor, sacrifice--and they would come away from these recitals understanding that they, too, would be expected to honor these ideals.
In addition, because all epics featured the intervention of the gods in the affairs of men, these poems helped the audience understand why certain things happen as they do. What might appear to be unexpected or illogical in a certain context, is understood to be a sign of a god's interference or intercession, depending on whether the result is good or bad for the society. This was a very important element woven into every Greek epic.