IMHO in my experience (mostly with wooden cups for Grado-like headphones) the material a cup is made of matters only to a certain extent, and only when looked at in combination with the design of the cup, because both of these elements determine how “active” the cup and the material itself is in the entire headphone assembly, or how much it will contribute to the sound. In my experience, making the parts “rigid and inert enough” that they don’t get significantly “excited” by the driver/sound is the point beyond which it doesn’t matter what material the parts are made of anymore, given that their design elements that can influence the sound (such as cup volume, opening sizes, cup shapes, internal wall textures, etc.) are identical. At that point, the parts are passive. Most headphones that do use wood in construction are pretty massive, and well within that area where the wood itself is essentially just a passive element. Some aren’t, like some closed-back cups that are purposefully designed with relatively thin walls and to not be be perfectly inert, which is why the material and the design of a certain part always have to be talked about in combination. The thing is, when you start depending on the properties of the material itself and the effects it has on the sound of a headphone, you’re sort of entering a territory where you don’t have perfect control over it, and you’re sort of playing trial-and-error games. Sometimes the effect of a certain wood/design combination will give you pleasant effects, sometimes it won’t. For that reason, I’m not sure whether the benefits of using wood nowadays outweigh the limitations of design introduced when dealing with wood. Especially in this day and age when various advanced polymers with various reinforcements are available.
If you’re talking about relatively thin parts of headphones, like a big shell of a closed-back full sized headphone with relatively thin walls, then obviously the material it’s made of will have a certain effect on the acoustics of that cup. But if we assume the part is massive and rigid enough, which can be achieved both through use of very rigid materials (like for example carbon fiber reinforced polyamide that I use in some of my headphone builds), as well as certain design elements, such as internal bracing or similar reinforcement structures, then it would matter far less what material the cup is made of, because it would be inert enough anyway. To give you an extreme example, if you were to couple a driver to a 10 pound heavy ear cup made of concrete, and then to a 10 pound heavy ear cup made of iron, both with identical internal cup volumes and shapes, would it really make a difference? Would the sound of those headphone be any different because of different materials? It’s an extreme example, but it’s just to show the direction in which my argument is going, which is simply that, given the right part design for the material it’s combined with, the material sort of becomes secondary, and you could even say it just becomes a matter of aesthetics.
As for the comparisons of headphones to wooden instruments, the issue is that wood in such instruments is active, whereas the wood on headphones is mostly passive, preferably at least. Take a guitar for example. It’s a relatively speaking massive body made out of really, really thin wood, with a huge internal resonant volume relative to the amount / mass of wood used. That wood really does resonate audibly and it’s an active part of that instrument. On the other hand, a wooden cup isn’t an active part of a headphone in most cases, and even when the wood is purposefully made to be an active part, it’s active to an infinitely lesser extent than it is on instruments.