That's right, you can't go missing. You can go fishing. Fishing is something people do. But you can't go missing. Have you ever tried to miss? How is it done? Perhaps if you are a man looking for a woman you could actually go missing. On the other hand if you are a woman do you go mistering? Ridiculous, you say? You bet. Our English language is going down the sewer.
You could, if you wish, simply 'vanish'. Or better yet 'disappear'. Or even 'get lost'. But 'go missing'? I rather think not.
If it were, indeed, possible to 'go missing', then it stands to reason that one could also 'go finding'. You see the logic?
Do you use this expression? Why? Because everyone else does? Perhaps you heard it on the news and thought it was cute. Then I say to you get lost.
P.S. The photo I had intended to place in the upper left corner of this post went missing.