No-- That isnt it at all! It isn't that elves dont like wood-- It's the exact opposite, coupled with an absurd dose of neurosis!
See, Elves have this "Self loathing" neurotic complex:
They are sentient, industrious beings, who try so very hard not to be, and suffer self-loathing as a consequence.
They revere the instinctual, almost autonomous characteristics of nature and natural systems as high prime ideals, which is why trees are seen as the ultimate expression of perfection. They employ a magical/special means of wood collection/crafting, because they HATE industrial processes, because they are "unnatural"-- Nevermind the cogitative dissonance that is required to hold this viewpoint, however. (The actual development of this magical methodology is itself a technological innovation, and is itself unnatural. I seriously doubt that any tree anywhere is going to start magically sprouting kitchen utensils and whole pieces of wooden armor all on its own. Sorry elves, that just isnt natural. Telling yourselves it is does not make it so. If you really want to think about the actual consequences of this development, you went from cutting off limbs from the tree, which is known to actually improve tree health in many circumstances when performed sensibly-- see for instance, Pruning, which simulates natural limb loss in a more controlled manner-- to actually ENSLAVING the tree, forcing it to grow in unnatural ways which we CAN PROVE it would NOT do on its own, just so you can feel better about yourselves. Which of us is the monster here again?)
Because trees are their mental model of perfect natural beauty, anything fashioned from them by "Crude industrial processes" is an anathema. It isn't that they don't value wood-- it's the absolute opposite-- wood is too holy and sacred to DARE be used. Instead, they choose to self-punish themselves with goods they consider to be of inferior value, as self-chastisement of their requirements for unnatural utensils and objects. "I hate that I need tools to survive, so I will outfit myself with ones made of materials I find less valuable in a futile and neurotic act of catharsis! That'll teach me to have to rely on tools!"
Basically, Elves are unabashed neurotics with an unrealistic view of what nature actually is, seeking to deny their own actual natures to be more like this unreal and projected ideal, engaging in self-destructive, irrational, and neurotic behaviors and (mental gymnastics to justify them), and getting bent out of shape when other races don't do the same thing.
Goblins are FAR more "Natural" than Elves ever even DREAMED of being.