Whom did the White Council serve?

[Ancalagon'sFire]

But at length the Shadow returned and its power increased; and in that time was first made the Council of the Wise that is called the White Council, and therein were Elrond and Galadriel and Círdan, and other lords of the Eldar, and with them were Mithrandir and Curunír. And Curunír (that was Saruman the White) was chosen to be their chief, for he had most studied the devices of Sauron of old. Galadriel indeed had wished that Mithrandir should be the Lead of the Council, and Saruman begrudged them that, for his pride and desire of mastery was grown great; but Mithrandir refused the office, since he would have no ties and no allegiance, save to those who sent him, and he would abide in no place nor be subject to any summons. from The Silmarillion

It would seem from the apparent make-up of The White Council that is was not fully representative of all the people of Middle-Earth, in particular it excluded those of the Races of Men and Dwarves.
Certainly we know Mithrandir and Curunír both sat on the Council, and although in appearance they were similar to Men, it was known that they not 'of' Men.

Even as the first shadows were felt in Mirkwood there appeared in the west of Middle-earth the Istari, whom Men called the Wizards. None knew at that time whence they were, save Círdan of the Havens, and only to Elrond and to Galadriel did he reveal that they came over the Sea. But afterwards it was said among the Elves that they were messengers sent by the Lords of the West to contest the power of Sauron, if he should arise again, and to move Elves and Men and all living things of good will to valiant deeds. In the likeness of Men they appeared, old but vigorous, and they changed little with the years, and aged but slowly, though great cares lay on them; great wisdom they had, and many powers of mind and hand.from The Silmarillion

If one considers that Dwarves still flourished and were patently opposed to Sauron, as were the realms of Men that stood firm against this evil, why then does this Council of the Wise not seek further representation from these races? Surely the decisions they took, the matters they discussed and the knowledge they collectively had would have been better served in their purpose had they included the leaders of these other peoples of Middle-Earth!