Where is "missing"?
Missing is a place that is not here and not known, I guess?
I would say, "She is missing", but I have heard "went missing" many times.
I think that maybe sometimes it could be used like a slang phrase.
Such as when educated people say "ain't" to affect the tone of their speech.
I suspect, though, that many people say "went missing" naturally and not intending to be affected.
I guess that it has become yet another one of those colloquialisms or vernacularisms that any colours every language.
The USA is filled with people who emigrated to there from every part of the world. Their native non-English languages affected their use of English syntax in many ways and their descendants inherited these interesting but skewed ways of speaking.
The New York City area is especially filled with unusual jargon and inverted syntax as a result of this. Then it ends up getting picked up as the local dialect by everyone in the city. Then it ends up as dialogue in tv and movies. Then people all over the world who are learning English from entertainment media pick it up too.
Such is the evolution of language. I guess. :p