Hey, they don't call it The Epic of Gilgamesh for nothing. Except for the early days of Enkidu in the woods and Utanapishtim's story of the Flood, Gilgamesh appears in practically every scene of the poem. Another thing that makes Gilgamesh the protagonist is that, pretty consistently, he's the driving force behind the story.
It's Gilgamesh's rotten behavior that makes the citizens of Uruk pray to the gods for help, which ends up leading to the creation of Enkidu. It's Gilgamesh's bright idea to go fight Humbaba. It's Gilgamesh's rejection of Ishtar that inspires her to send the Bull of Heaven down to earth, and sets in motion the chain of events that leads to Enkidu's death. And it's Gilgamesh's grief over Enkidu that inspires him to go on his great journey beyond the ends of the earth.
For all these reasons, we think you'd be hard-pressed to find a more classic protagonist than Gilgamesh.