Good question.

His desire to become the champion came about after he heard about Red. For him, it's like the ultimate expression of proving his existence. While Roxas has developed enough to know that he does deserve to exist and belong in the world, becoming the Champion would be like the icing on the cake. Roxas does have a competative side to him, after all, as seen in the struggle tournament, and I think he does like being on top. Also, I think after seeing other trainers, both Route characters and NPCs, going for the badge challenge, he accepts that as a norm of this world.

That being said, Roxas isn't in any rush to get all the badges and fight the Elite four and all that (after all it took him two years to get all the Johto badges). He's content with travelling around and seeing the sites and catching pokemon.

I admit it's also a device for me to prevent stagnency and provide some motivation for Roxas and what he's doing in Johto.

So yeah, I hope that made sense and I managed to clarify that for you, anon. Thanks for the question.