I never got the “wipe the sweat off sick people”. Just rinse yourself off in the bathroom. Japanese bathrooms have hand showers, and in the worst case you can just dump some water in yourself from a basin.
@Guspaz probably because people used to go to public baths instead of having one at home, which would naturally lead to the idea of "bathing during a cold is detrimental".
Gotta be patient until the manga universes (and their creators) can break free of cliches and adopt the latest medical advice
also going to the bathroom when you have a fever is kinda annoying ngl