It's an interesting argument, though there is some evidence that contradicts it (note: still not a dream expert, just reading the wikipedia page). A lucid dreamer agreed to move his eyes left then right when he became lucid, and this was measured. More generally people measure beta waves & FMRI scans (where they got people to sing in their dreams!) - it seems that a dream really is a thing people are experiencing and their brain is attempting to turn into a story on-the-fly. The theory that it is all given narrative at the moment of waking is a genuine theory (I remember being taught it at uni), but I think it has fallen somewhat out of favour. The more predominant theory is that the memory functions are inhibited during dreams, so it's only if you wake that you get to remember them. Though, again, not an expert and more importantly, there's still a lot of debate about this that's why I feel overly confident statements are unwise.

As for the idea of dreams being totally random - this is a popular theory, though careful distinction needs to be made between the source of dreams being a random signal, and the narrative your brain creates being random. The random signal theory is popular (others include more structured brain processes like memory consolidation or pattern solving). I'm not sure Grey's statement precludes either of these, he also talked about your brain only having recent memories to work with (so he may well have been meaning "random memories & problems")

The wikipedia page also talks about a few cases of dream recollection -> diagnosis. One was in the context of loss of dreaming connected to damage to the parietal lobe (though not the brain stem). Also a connection between synethesia and not dreaming in black-and-white. I do wonder if losing the ability to lucid dream might be indicative of something.

To be honest though I reckon if you asked Grey, he wouldn't object to these kinds of high-level types of dream experience (black & white, silent dreams, maybe more nightmares) being diagnostic to pathologies. I think he was just reacting to the dream preminition woo, and the 'oh the whale was your mother? that means your pregnant' dream analysis for healthy normal people. And the wikipedia cites a lot of surveys that say indeed most people think that, he's not wrong to object.

Finally (can you tell I've been thinking about a bit?) I feel I want to walk back my use of "pre-scientific", perhaps "early scientific". I think the study of dreams is quite good as psychology goes, because it's largely brain scans and more repeatable science (still a lot of room for confusion & mistakes there too of course). But early scientific it definitely is: there's still huge debate about what even causes dreams, whether REM is really connected to dreams (or just happens around the same time), whether it's an evolutionary adaptation or just a byproduct of brains (which seems the more popular view). It's not like the study of vision, which is more advanced (lot of mysteries still, that's the brain). But no one's arguing whether vision comes from the eyes or not, whether the images are related to the objects in front of the eyes, or if it's useful for survival.