I wonder what you mean by saying there wasn't anything like anxiety as it's experienced today in the premodern world. The term "anxiety" comes from Latin "anxietas", and Old English had "angsumnes", both with apparently similar meanings, so it seems the idea has been around for a while, and most likely because it's a basic human emotional condition that could occur in various circumstances throughout history and prehistory.
As for a relationship to death, it's difficult to generalise over all people throughout history, but for example in medieval Europe where you had a seemingly quite vivid belief in the possibility of hell after life, that could result in some extreme anxiety of the most inescapable kind. Maybe that accounts for some of the more extreme behaviours of those times, who knows.
