Have you noticed this thing where the most famous Western epic, The Iliad, is really non-standard as an epic? Meaning, an epic is generally about a man who embodies cultural virtues and acts them out, typically in combat or adventure. Odysseus, Rama, Beowulf, etc. But the Iliad is centered on a guy who is almost culturally perfect except has a tragic flaw of wrath.
Perhaps it'd be better to think of epic more broadly as a story about cultural values, which may feature a Beowulf-ish character.
@ZachWeinersmith I feel like the Iliad is working through different heroes and how they each embody different ideals of ancient Greek culture. Achilles is divinely powerful but driven by passions. Odysseus is clever but untrustworthy. Agamemnon is a great king but not himself much of a warrior. Ajax is an ordinary guy trying to be the best hero he can be among demigods. Hector is noble but sworn to protect the enemy city, etc. Each represents their own way to be a hero, a different ideal person.