> How long do you sit in a bath?

Another Japan resident here! It is common for a family to share a bath for the evening. (One at a time!) This works because each person showers to get clean before you get in the bath. Many also have folding bath covers, which help to avoid the water from cooling down and to keep the water clean during the showers. Still, nobody wants to be last.

> My immediate reaction to this is, “there’s a way for my house to be remotely flooded”.

A smart reaction! But most baths are also whole-room units, with a floor drain, and are intended to get wet. I myself have forgotten to stop the bath water a number of times (neither I nor my bath are so "smart") and while it is an unfortunate waste of water, there is no danger of flooding, only slight embarrassment.