For me, I feel like I prefer a class to describe what the characters does -- a fighter fights, for example. Not all classes do that well, but barbarian probably does it the least. A barbarian rages/berserks, pretty much, in D&D. But you could totally have a civilized elf battlerager type.
I feel like I have to poke at this

. You say barbarian does it the least? I'd say it does it as well as most of the classes. Few of the classes actually describe anything resembling what the character does. They describe a noun, and we all bring our shared assumptions about the verbs we associate with that noun but which are often unrelated to the noun. And we roll with it.
Fighters fight. I'll give you that one.
Bards... bard? It's a derogatory term for an itinerant musician, and so far as I know there isn't a verb form of the word.
Monks... monk? Again, there isn't really a verb form.
Rogues... rogue?
Barbarians barb. That is, they are barbers. Obviously. This is the second most clear name, right behind fighter.
Clerics clarify.
Druids druify.
Paladins pallad.
Rangers range.
Sorcerers ensorcell.
Warlocks despise locks to such an extent they have declared war on them.
Wizards wiz.
I'm not saying there's no virtue in wanting class names that describe what they do. Just that 5e doesn't do this with any amount of intentionality and barbarian is definitely not on the bottom of the list. I'd be all for classes that did, though. Fighter, healer, spellcaster, etc. I feel like it could really help consolidate the ballooning list of classes and would help bring a nice level of focus to the game design.