Try secular non-fiction to exclude both myth (fiction) and religion.
1 Not connected with religious or spiritual matters
Prose writing that is informative or factual rather than fictional
Your sample sentence is then
While it seems impossible to have a story without a connection to mythology or religion, some believe that secular non-fiction is an exception.
You have taken pains to separate the philosophical discussion from the English phrase request. The discussions on this page have led to the philosophical position that stories that are merely motivated by religious themes, such as your example of the triumphant underdog, are to be labelled religious; and stories that could be given a religious interpretation even after they have been written are also to be labelled religious. By definition, then, those stories are not secular.
The question is not about whether it can exist, but is there a phrase to describe this situation.
This answer is compatible with the notion that although philosophically there may be no such thing as secular non-fiction, that's what it would be called if it existed. That is, although I offer a name for the category (the English phrase to describe this situation), I don't claim that any story exists within that category (cf The question is not about whether it can exist).