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.