What it really comes down to is that the expression, "You'll catch a cold", is just a general shorthand for "That's not healthy for you." It's not about not understanding germ theory — really, Japan learned that in the Meiji period, not very long after Europe did — it's about expressing concern.
(And I'd mention that in English, we still call the upper respiratory infection by any of a number of rhinoviruses, influenza viruses, or human coronaviruses a "cold", on the same reasoning as Japan uses— and that there's scientific evidence that hypothermia can make you more susceptible to such an infection.)
Very cute story — I particularly enjoyed Kanan's bit about waking the other two. Adorable.