So, I can’t honestly personally understand taking your shoes off in an emergency, however I am sure they have gotten very efficient at it.. I can say that if they didn’t it would cause damage to the home, many Japanese homes, especially traditional but also in general just aren’t built to take shoes. Tatami mats, and even hinoki floors despite being “hard” for a softwood, is still a softwood, so some level of genuine damage would occur. Again of-course versus the emergency in saving a human life, I’m not quite sure I can understand it.
Most shoes have rubber soles, to do damage to the floor it would have to be softer than rubber.