I'd say that you're likely thinking too much of caves and not enough of real-life underground environments like one of those forest-caverns in the Amazon—it would functionally just be a dark forest zone like Drustvar, albeit with no visible sky when you look up and bigger spiders. I could see a very interesting zone out of that—a sprawling, dark underground forest like an evil version of Sholazar Basin covered in webs. I also always envisioned a second underground forest beneath the Grizzly Hills full of mutated creatures and with Emerald Nightmare-based aesthetics. A frozen cavern alone could help to provide variety, akin to Naz'Anak in WotLK. Those are two-and-a-half examples of zones that I think could be very good if they had some variety.
I think that what is being ignored here is a great deal of zone variety that could be done to keep it lively. I understand what you mean, of course—too much in the way of dark, gloomy zones would be rather grating, and I could see it feeling claustrophobic if the roof is too low, so I'm not wholly refusing your premise, but I do think that all it would take it a very well-handled development. I've made a few expansion concept maps from time-to-time, and I've done a couple of Azjol-Nerub-style maps—some as actual cavern systems, others as caldera-esque expanses like Deepholm. I think the issue of the gloom becoming banal could also be solved, such as with a Naz'Anak zone that's bright and colorful due to the ice walls and bioluminescent life, leaving claustrophobia and the hidden sky as the only things that may come off as alienating or oppressive. Admittedly, it is hard to come up with good concepts to deal with that, as even a very high ceiling could still cause those problems.