I really love how Roku and Sozin’s story is almost an exact mirror of Aang and Zuko’s - everything that happens in Roku and Sozin’s story happens in Aang and Zuko’s, but in opposite order. 

A hundred years ago, the world started off in an era of peace and balance.  Roku and Sozin grew up as best friends, but then Sozin began to lose his way and go corrupt with power, beginning the Hundred Year War.  This created a rift between Roku and Sozin, and they became bitter enemies to the point where Sozin simply watched as Roku died.  Fast forward a hundred years, and we see in “The Boy in the Iceberg” that Zuko watches as Aang is freed from the iceberg (and thus kind of reborn in a way).  Both characters were bitter enemies from the start, and the rift between Avatar and (future) Fire Lord continued.  Yet as Iroh points out in “The Avatar and the Fire Lord,” “what happened generations ago can be resolved now, by you. [...]  Born in you, along with all the strife, is the power to restore balance to the world”:

And Zuko does restore balance!  He finds his way (in contrast to Sozin, who lost his way) and lets go of the corruption thrust upon him by his forefathers.  This closes the rift between Avatar and Fire Lord, ends the Hundred Year War, and brings the world back into an era of peace and balance.

So, by forging his own destiny, Zuko not only became the glorious Fire Lord he deserves to be, but he also brought the legacy of the relation between Avatar and Fire Lord full circle, fixing all his ancestors’ wrongdoings and resolving the conflict within himself forever.

The scene in “The Beach” where Zuko furiously yells that he’s angry with himself reminded me of a similar scene in “The Storm” with Aang:

-Both characters aggravate a campfire and cause hot cinders to fly everywhere. -Both characters lose themselves for a moment because of their frustration at trying to figure out their identity -Both characters think a lot about their past in relation to their current identity (Zuko took a walk through his family’s old house on Ember Island and had brief flashbacks)