I have always thought the word went indicated, more or less, something purposely done; yet, it is now used when there is no idea how something disappeared.
One meaning of the word 'go' (of which 'went' is the past tense) is:
to pass into a specified state
Thus, it acts similarly to the word 'become' and in that case, works perfectly well in the example cited in the question.
He went missing. vs. He became lost.
Hope that helps.
N.B. Understand then, that in this case, the word 'go/went' is intransitive and is nothing to do with movement as it would be in the sentence, 'He went to work."
There is no connotation of "deliberately" that attaches to the verb "go" (or its past tense "went").
In the last sentence, "went missing" just means they are no longer in sight. Perhaps the wandered off, or the speaker got lost, or whatever, but no purpose is implied by the statement.
My answer is based on that topic, from Ask the English teacher.
“Went missing” means “disappeared”, or “someone or something didn’t show up as expected”. Metaphorically, it can describe someone not paying attention or failing to attend to a task.
According to the following n-gram, the expression has been around for some time, and being used more and more since the 1970’s.
Also, here’s a citation from the post I mentioned on the beginning of this answer:
Based on that, i.e., the expression has a stable, well-defined meaning, it’s been used for decades now, and a native speaker think of it as “normal language”, I can say yes, the term is correct.