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!

[Inziladun]

The White Council looks to be a gathering of those who saw the 'big picture' in Middle-earth; those with the most perspective and long experience. The Dwarves were always guilty (save in extreme necessity such as the Nirnaeth Arnoediad against Morgoth, or the Last Alliance) of being very self-involved. I can't picture the Dwarves of Erebor taking the time away from their own affairs to recognise the importance of council with other powers in Middle-earth. It was only immediately before the start of the War of the Ring that they were forced to do so.
The same could be said of the Dúnedain, at least those of Gondor in the Third Age. Boromir, for instance, thought only Gondor stood as a barrier to evil in the West, and its only allies were the Rohirrim. Gondor had come to the point where they believed they had no other allies, and it was only Boromir's dream and Denethor's anxiety about Mordor that brought a high Man of Gondor to the Council of Elrond.
As for the Dúnedain in Arnor, they would have been too busy keeping guard, and Aragorn probably wouldn't have thought himself deserving of a seat on the Council anyway.
The Council had to consist of those who could govern the 'real' war against Sauron without thought of any individual conflicts or interests. The greatest remaining Eldar and the Istari seem to be the closest to that ideal.

[Formendacil]

I think it's important to note that the White Council was not exactly a legislative body so much as a group of like-minded people who got together to plan strategies against Sauron. In line with what Inziladun is saying, I'd say that this like-minded group really only consisted of the Istari and the Eldar.

In the time of the Last Alliance, the Dúnedain would probably have belonged as well--but by the time the Istari showed up and the White Council formed, Arnor had declined into petty infighting and Gondor was myopically swallowing up the southern coasts--neither would have seen the big picture at all. Perhaps later, after the fall of Arthedain, the Dúnedain of the North would have been involved in the actions of the White Council, as allies and collaborators of Rivendell and Gandalf (ie. guarding the Shire), but I can't really see any of the rugged Lords of the Rangers sitting in council with them....

[Morthoron]

And let's not forget that the immortal make-up of the Council could allow for planning over centuries, rather than just months or years. Having mortals on the Council, even longer lived races like Dwarves would reduce the overall effectiveness of the group, particularly among Men, whose political and philosphical changeability might be pronounced over several generations.