Chihayafuru is focused on both: episode 13 mostly eschewed karuta for character development, while episode 14 was all about the game. Be careful that you don't try too hard to simplify things for the sake of easy classification.
Personally, I see Chihayafuru, barring the "prequel" episodes, as more like Giant Killing or OoFuri than Cross Game.
I personally see the karuta of Chihaya as a vehicle for both character development and thematic transformations, and not much more than that. I'm not seeing how they can be separated, really. Hosanna, I believe even you made the mistake of thinking that ep 14 was all about the game, when it was just for the larger purpose of bringing The Queen into perspective.