At first reading, Beowulf seems to be an exemplary hero. Once you have started examining exemplary heroes around the world, however, you begin to notice that the texts nearly always sneak in some criticisms. Odysseus is a first-class example of a survivor, but not even a third-class example of much else. Gilgamesh is in his personal life a mere rapist, partially subdued and reformed by Enkidu - and then the two of them go off on a bro-adventure. I am not sure we would be so attuned to tragic flaws leading to tragic heroes if it weren't for Shakespeare and a few Greeks, but now that we have seen it, we can't unsee it. The manuscript is clear that Beowulf is a great hero, and even in his 70s is the only hero who can go against the dragon. Yet it hits this theme long enough, and wrings its hands over what will come next that the reader/listener has to consider why there is no one to come after. Has Beowulf in fact failed to raise up a generation of leaders and heroes after himself? There is only Wiglaf, and his nobility seems to derive from blood and intention, not training.
Unrelatedly, Beowulf is put forth as the national epic of England. I disagree. I don't much mind that he is ambiguous as to membership in Fair Albion. He is a Geat (pronounced "Yat,"), but that is pretty typical for all heroes. Among their many ambiguities, heroes of the great epics often come from border categories of the nationality in question. Still, he is pretty far afield even by the loose standards of epics. I would nominate the general character of Arthur across the contradictory literary works about him instead. There has been a shying away from Arthur over the last century because his actual historicity seems dodgy. Why that would point us to Beowulf instead is a bit humorous. But no one in 1900, 1800, 1700, in England would have picked Beowulf as any representative of anything. Most would not have heard of him, though Arthur would be recognisable. No, Beowulf has risen up because introductory British Lit courses have started there for the last 70 years, imbuing the work with an aura of UK continuity that is not there. Also, there are a half-dozen texts about Arthur that could be put forward, none commanding the heights, so they don't jump off a syllabus.
Still, Arthur is the one. Despite all the arguments and confusion, he is where the heart goes.