I thought AFFC was relevant to really introduce the other players affecting the story's outcome. These are the Greyjoys (Euron, Victarion, Asha, and Aeron, hmmmm, it just occurred to me that Aaron, Moses' brother, was also a younger brother and prophet), the Martells (Doran, Arianne, and the Sand Snakes), and the Faith (the sparrows and the militant orders).
Perhaps the main point of AFFC was to continue the development of Jaime Lannister. If AFFC or ADWD had just picked up five years after the Red Wedding, Joffrey's death, and the fall of Tywin Lannister, would we have even recognized the changed man Jaime will become? I, for one, would probably have blamed GRRM for not filling us in on how Jaime did a 180.
After Jaime, Cersei's development, or should I say devolution, was germane to the Iron Throne. Cersei is evil and she could have been used to introduce more evil characters, but her descent into foolishness and the introduction of her fools was fairly interesting. Joff's petulance and Cersei's debauchery are there to recall the totally amoral reign of Aerys II.
Also moving smaller plots along were Arya, Sam, Sansa, and Brienne. Arya's story is her own personal story, imo. Sam's story is being used to set up the return of magic, the return of Dragons, and the introductions of Rh'llor and The Other. Sansa's story is the story of the Starks. Brienne's story and her subsequent death is the author's reminder to not get used to supposedly central characters. GRRM likes to lead us down dead ends and/or set up road blocks.
In the ninth chapter of AGOT, Jaime is shown to be in an incestuous relationship with his sister and tries to murder a child. I never thought that he'd become my favorite character in this series. For me, the Lannister brothers are the two most believable, enthralling, and thought-provoking characters by far. Even though AFFC did not have Tyrion, I still enjoyed it due to Jaime's soul searching, his repentance (though he doesn't yet admit it to himself), his newfound graciousness, and his ruthlessness.