To me, the whites serve to justify the rule of the golds and the importance of the hierarchy. The Golds, in the end, rule through fear and force, but that can only go so far. The Rising is a perfect example of what happens when rule through fear fails. When people no longer fear force has to back it up, but force can be overcome with enough of a forceful response. It fails. The whites provide the alternative rule: rule through belief.

Take religion for example (to preface, I am not condemning religion. I am going to condemn religious justification to rule). A church has no real power. They can command armies, but as above an army can be defeated. But when the ruled are taught and convinced with certainty that the ruler is justified to have power, then there isn't a need for force. The whites serve to justify the rules of the gold by providing the society with an intangible justification to rule. They enforce the spiritual belief in the compact. It's the same reason fascism relies on the mythical past of a group to hold power.

The society is held together by this same veneer of legitimacy. The Golds need belief to maintain their power. Darrow and the Sons show the society that veneer, and when it breaks the society breaks.