I agree that go missing is not the same construction seen in go walking or go fishing, and that missing here is acting as an adjective.
But you know, I am not completely buying AHD’s “Go fishing involves a different sense of go: transitive sense 3 “to engage in” “…
It seems a bit forced to say that there is a construction in which go is a transitive verb, but the object has to be a gerund. To me it _feels_ like a phrase involving two verbs. To me, “He’s gone fishing” means more than “He’s engaging in fishing.” It means he has gone, and he is fishing. You have to depart, then do verb 2.
I mean the construction is only used when someone has _gone_ in some sense. If Bruce is standing right next to me, engaged in baking, and someone on the phone asks me what Bruce is up to, I’m not going to say “He’s gone baking”.
It’s a bit akin (not completely analogous) to “Go jump in the lake!”, “Go fly a kite!”. I’m telling you to do to things: get the hell out of here, and then do (2nd verb).
I realise this isn’t a very solid case, but it’s the vibe, man.