I'm currently mitigating those risks. The bucket name is not enumerable/guessable, even with brute force tools, and the filenames aren't predictable either. I have bucket enumeration set to forbidden.
What would be the value? If my family members send a link to someone else, that person will have the link forever? If the unauthorized person has a link to a video, they can simply download that video and keep it forever anyway. They can't retrieve other videos in the bucket unless they can correctly guess their (not very predictable) filenames.