Is democracy really worse than random?

That is indeed what systematically biased voters imply. Because so many people vote, the incentive for any one to correct their bias is negligible -- the overall result of the vote is not affected by doing so. Also consider that an "everyone votes" system has the expense of the vote itself and the campaigns.

Probably because that word doesn't mean what I think it means.

Ok, it wasn't clear that you were talking about voting within a republic from the initial post.