Assuming that they keep a lot of the previous Phoenix Clan dynamics, it would seem to me that the Shiba would be the most able family to fill both the military and political aspirations of the Clan. The Asako family's on internal lore made their place as the clan's "face" was a bit at odds with their ultra secretive super monk focus. In addition the old lore had the Isawa just this side of hostile towards the Asako (due to jealousy of Shinsei's favor). That makes it seem unlikely to me that they Isawa would want to place the entire clan's courtly glory upon a family they look down on and would like to see humbled. Then we have the Isawa themselves. Due to their tendency to only be concerned with the kami, "pacifism", and maho they really have never had a real grasp on the realities of the Empire.
With the above considered, that leaves us with the Shiba. The Shiba keep the peace between the two spiritual families, they keep their eye on the external threats to the Clan. They also are more service oriented. Finally they're the actual "royal" line of the Clan. While the Shiba like their founder are humble, the rest of Rokugan respects their lineage and sees them as the Clan's "true" Champions. This would, in my view give them the edge on courtly behavior, and honestly I think it's appropriate. So the Shiba represent and protect the Clan on battlefields both political and military, the Isawa seek the Clan's favor with the kami and fortunes, while the Agasha follow the disciplines of the arts and spiritual enlightenment.
The Dragon should keep the Agasha, that storyline went south rather quickly and I'd like to avoid those mistakes. As I see it the Phoenix has the families it needs to get the job done.