Regarding the D&D tradition that causes this confusion in 5e, it is an incoherent tradition. The Druid is an elementalist, who masters earth magic, including metal magic, and uses metal weapons proficiently. Metal tools too. To be unwilling to use metal armor, but be great at metal magic and metal weapons, is inexplicable.
Indeed, the words "will not" are a conspicuous absence of the flavor that is necessary to explain, why not?
Probably, the game works better (at least more coherently) to drop the metal prohibition from the Druid proficiencies.
Part of the problem is the way 5e delays the choice of the specialization archetype until a higher level for some classes. For the Druid class, the choice of circle defers until level 2. At least, I would enjoy the game more if every class had some way to choose the archetype at level 1.
For the Cleric class, the choice of domain happens at level 1, and different domains grant different armor proficiencies. The same would work well for the Druid class. One circle that focuses on animals and beast magic might only gain proficiencies for armor made from animals. But an other circle that focuses on the earth element might even require only metal armor. And so on.
If the armor proficiency is called out thematically, should the same theme apply to weapon proficiency too? Maybe, the metal-shirking Druids should only use weapons of bone, horn, and ivory? Like the Shillelagh cantrip for a wood club, maybe the Druid can have other cantrips to wield these beastly weapons magically.
If there is a mechanic − especially an unusual mechanic − there needs to be a flavor to go with it to make sense of it.
And this flavor needs to be flexible, and adaptable, to use for different kinds of character concepts in different kinds of settings. By itself, the prohibition against metal armor (and only the armor) is too peculiar to be useful for other settings and other character concepts.