No.
Even if every single politician disappeared, we would end up with new ones, whether or not our government was elected or appointed by the Great and Powerful Oz. Even small tribes needed people with political skills to keep things working, to encourage folks to get along and to keep from running afoul the neighbors. As societies became more unified, as tribes merged into larger and larger communities until there were nations and empires, the need for leaders with political skills increased.
The problem is not that we have politicians, but that we often expect something of them that they cannot deliver or that we allow ourselves to be duped.
Your comment about small tribes got me thinking. It's totally true that "leaders" always come up in any group. That's one of the reasons that I usually dislike being in groups of 3 or more people my age (I think it's different with children). If anybody has tried traveling with 3 of people people, you know how horrible it can be. It takes forever to make any decisions and there's usually somebody unhappy with the decision. You're always waiting for somebody. The only way that it really works is if somebody takes the leader. Honestly, they have to pay a little bit less attention to what everybody wants and just make decisions, so that something will happen.