That covers the germs and steel. Where do the guns factor in?

Minor addition:

In Europe, to make drinks like beer you had to boil the mash, which unknown to them sterilized the water, which made beer generally safe to drink.

In east asia, as you mentioned tea was a common drink. But before that there were numerous herbal remedies that had to be boiled and served hot as well. People who drank the herbal remedies got better (mainly because hydration and clean drinking water are important factors for well being). Other than attributing the recovery to just the herbs, they also attributed it to the temperature.

So lacking tea or herbal drinks, the ancient chinese believed drinking hot water was somehow beneficial to the body. Add that to the fact that many who drank cold untreated water fell sick, you can easily see how the myth developed.

Another side note. Hot water is expensive (fuel wise) so drinking hot water was a sign your family was comparatively well to do and something a lot of villagers emulated in an attempt to show that the family was well off.

I know of two ways it can be. You know how the body fights infection with a fever - well I have some chronic inflammation in the gut that is exacerbated by cold exposure, and the gut becomes more leaky after going out in the winter. Tyramine from aged foods leaks into the bloodstream, causing various symptoms. I drink hot water now.

The other is
https://en.wikipedia.org/wiki/Idiopathic_cold_hemagglutinin_syndrome

Yes, body dampness. It's old Chinese medicine and it's believed cupping helps draw out body dampness. Among other things.

I love that kotatsu idea! That would be such a cozy way to rest in on a cold weekend. This should be a thing everywhere …. But only the electric version. I’m not sticking my feet under a mystery blanket with a charcoal burner somewhere