How exactly do you propose we say we don't know where someone or something has gone since the last time we had seen them?
Disappeared doesn't work because they more than likely still exist and didn't vanish while you were watching them.
Lost doesn't work because you don't have to be lost to be missing; you could have simply decided to leave.
I guess the best you could really do is "...has been missing since..." instead of "...went missing..."
Perhaps a more accurate description of what happened. Example: "The little, screaming brat was not where his mother left him when she returned for him at 3:00pm."
This whole "went (has gone, etc.) missing" nonsense is fairly new. I don't recall hearing it prior to, say, 2000. Certainly, for most of my 53 years, reporters were quite content to use the term "disappeared." Example: "When his mother returned at 3:00pm, the little brat had disappeared. The neighbors rejoyced."
The first definition of "disappear," in the American Heritage Dictionary of the English Language, Third Edition, is "To pass out of sight, vanish." Further, to quote from the "Synonyms" section for "disappear:" The central meaning of these words is "to pass out of sight or existance."" An example the editors give is "A skyscraper disappearing in the fog." A skyscraper vanishing in the fog certainly does not cease to exist, and there seems to be no requirement for an observer from which it disappears. There is an implication that IF
an observer was present that observer would notice the skyscraper disappearing from view.
What I am saying is that "disappear" works very well, and it is a much cleaner way of stating it than the horribly awkward "gone (went, etc.) missing."
And, for what it is worth, the phrase "went missing" is grammatically correct. The verb "go," of which "went" is the past tense can be used to describe a change of state: "My hair went grey several years ago." Really, it did. Started to go
grey when I was in my mid-twenties. See how well it works. And "missing" is a change of state, so it can be used with "go." But it is awkward and sounds stilted. The stilted sound or construction of "go missing," is emphasized by the, usually, very casual style of writing for news reporting in which the phrase is set. In that context, "disappeared" simply flows better, is not inaccurate, and sounds much better.