Oh, cool. The media says "supposedly" and "allegedly" to avoid trouble, but I guess it is different because even if you saw the dude shooting someone, you still have to call him an alleged killer. Because maybe he was a sleep walker, or the victim had asked him to shoot him because he was ill, or he did it in self-defense, or they were shooting a movie scene and nobody knew the gun had real bullets, etc.
You can't use them like that. That only works after you have used a word like everybody/somebody/nobody. Like: Everybody worries about their own life. There it's fine to use 'their' instead of 'his or her'. It's understood.
But you can't use 'they/them' etc in every context just because you don't know the gender.