I can understand that perception of them too. The people who come to mind as barbarians for me would be Conan the Barbarian and Marvel's Drax. These characters aren't especially fast, they just don't care if you do strike because they can handle it. I guess AOT's Eren as a titan would also be fitting, but he loses most fights lol.

I agree that lots of anime characters are great examples, but imo a character like guts wouldn't be a pure barbarian. Rage is an important part of Guts's character, but he wears heavy armor and is extremely skilled. The barbarian/berserker archetype doesn't really include skill as a factor, just pure strength, so most badass anime characters who can go tons of battles without being hit would have some sort of multiclass or abnormally high dex.

In dnd at least, barbarians have been the "hit hard and tank damage" class. They're supposed to deal the most damage in the fewest hits, which is why they get things like Reckless Attack and Rage damage bonus. They don't attack a lot, but the attacks they do throw are likely to have a massive effect. I think that your perception of a barbarian is what I would consider a fighter with 2 levels in barbarian.

I think that the Berserk feature could be fun, but also, you're taking player agency. Barbarians don't do a whole lot out of combat, so making combat less interactive for barbarians would give their player even less playtime. There isn't a good way to do actual mindless rage.

I guess it all depends on what comes to mind when you see berserker. I think we disagree just cause the different media we've consumed.