Is Beowulf the ideal epic hero? If not, what is he lacking?

In many ways, Beowulf is the archetypal epic hero, embodying much that was held to be good about a warrior in the Anglo-Saxon context. He is brave, has superhuman strength, and respects the kingly position of Heorot, taking on the king's greatest enemy alone as a means of honoring the king. In this, he is an excellent vassal, and, later, he grows to be an excellent king, dedicated to leading his people and improving their lives. The key example of Beowulf's bravery is, of course, the section in which he fights and defeats Grendel, which puts him in extreme personal danger, as a service to Heorot.
Beowulf's military strength, however, is inherently tied to his key weakness: his "ofermod," or over-confidence. We can detect the suggestion of this early in the poem, when he boasts of his previous exploits: heroic boasting is a key part of Anglo-Saxon warrior culture, but there is some indication that Beowulf exaggerates in order to build up his personal reputation. He may have superhuman strength, but is it really possible that he could hold his breath underwater for as long as he claims? Where a truly ideal hero would perform his feats selflessly, the use of the word "ofermod" in connection to Beowulf's decision to fight the dragon alone is telling: it suggests that the poet feels Beowulf is not pure in his motives. That is, Beowulf does want to save his people from the dragon, but he is also driven by the desire to increase his own reputation, and this need for personal reputation ultimately leads to his death.