I don't think the main thrust of your argument is wrong, however, your sickness example is also a false equivalence.

There's nothing logically or philosophically or scientifically preventing both explanations for why we each got sick from being simultaneously true. You're attempting to draw a dividing line between "religious thinking" and "rational, scientific thinking". No such division existed throughout history. Even today many "scientists" think along religious terms while many religions are evidence-obsessed. Counter-examples abound.