Caution, it is extremely insensitive to play the EEW alert tone in public, in Japan. You will receive a near-hostile response. Don't do it. Watch those videos in private, or with headphones.
Is it any more insensitive than watching any video in public without headphones? Myself I wouldn't ever do that with any video to begin with, so no need to do it for specific videos. Seems just like plain common courtesy.
Yes, there is a difference between playing random noise and playing the sound of an emergency warning system.