No, the poem isn't just about a bird; we can also interpret it as a comment on absolute power and the effects of such power on humans.  Personifying the eagle as having "crooked hands," gives us a clue that the eagle might be symbolic of a person, since people have hands.  Further, there are two definitions for crooked: bent (like an eagle's talons would be) and corrupt (as someone who holds absolute power likely is).   Perhaps this supreme ruler holds onto his power through corruption.   

Moreover, the eagle is "close to the sun," implying that his power is, indeed, absolute; he is the highest authority perhaps because he is the only authority.  The fact that the sea beneath him is described as "crawl[ing]" aids this interpretation because crawling is an action associated with someone who is powerless, subjected to the demands of a supreme authority.  Consider how we describe insects as crawling (when it might be more correct to say that they there are walking or scooting), and insects are typically at the mercy of larger predators because those predators are stronger than the insects.  To the eagle -- or the supreme ruler, those "beneath him" -- either literally or figuratively -- are as insignificant and unimpressive as bugs.  

Finally, when the narrator says that the eagle watches from his mountain walls, it is notable that mountains are natural, like the eagle, but walls are manufactured.  Thus, we might interpret the line as an eagle standing atop his crag, or a supreme leader ruling from his fortress, an edifice that protects him by keeping others out.  In the end, the eagle "falls" like a "thunderbolt": we could read this as an eagle diving for prey, quickly and destructively, or we could interpret it as the way a supreme leader tends to lose power -- all at once and violently.  When a supreme leader falls, it is usually the result of war or a coup, and so on.