Model Behavior
In Soul Society, it seems nobles are the typical fare in the shinigami academy, which means that the profession must be very highly esteemed. However, in the present time line, it seems that the Kuchiki clan is the only one of the highest ranking families to hold a captainship for at least the past two generations. This pillar of loyalty is a welcome constant in so changeable an environment, and Byakuya intends to hold that pillar steady.
Power Symbol
A behavior inherited from his family experience, even the proud Byakuya puts himself at the mercy of whatever the higher-ups in the Gotei 13 decide. As a captain and as a noble, he feels must be the champion of law and order regardless of whoever he must face to preserve them.
Unfortunately, this obsession with the big picture apparently leaves the captain blinded to what's beneath him--including the interests of his subordinates. This parallels his issue with familial duty, because he primarily concerns himself with what he can do for the system without an afterthought of what he can do for his division. A knight playing with his pawns seems like an appropriate image, here.
As the story develops, the continued loyalty of his Lieutenant Renji seems to indicate an improvement in Byakuya's general care for the division, though there is no concrete evidence of that shown.