Can a Clan remain relevant and powerful even if only one member remains?

In many fanfics, there's a Council in Konoha, made up of the Hokage, his advisors, and the leaders of the most important clans in Konoha. I was wondering if anyone could still represent their clan on this Council, and if they could still have the influence they had when there were more of them.

Like Naruto, the last Uzumaki in Konoha, who must have had a lot of political significance at the time given his close ties to the village; Sasuke, the last Uchiha in the Leaf after the Massacre; or Kakashi, the last of the Hatake (who, to this day, I'm still not sure if it's a real clan, but let's say it is).