Given that the description of a class is just as immutable a rule as its mechanics - which is to say, it isn't at all, but all changes are subject to DM discretion and approval - it is just as much against the rules for a barbarian to focus on a weapon inappropriate to their upbringing as it would be for them to wear armor they weren't proficient in. That is to say, you should definitely talk to your DM about it, because it's very weird.
Since at least AD&D 1E, there's been a problem with some classes being over-defined and others being under-defined, in terms of conceptual narrative space. In the Basic game, whether you want to play an acrobat or an assassin, you can do that as a Thief. If you're playing AD&D with Unearthed Arcana, your acrobat should probably be a Thief-Acrobat and your assassin should really probably be an Assassin.
The same general problem got much worse under 3E, and I'm not just talking about the Samurai class which took the last remaining archetype from the Fighter. It became a major issue if you were using all of the supplements (as many 3E fans were inclined to do), that any cool thing you might want to do was locked behind some feat chain somewhere. If you were just using the PHB, you could try and fire an arrow in such a way as to pin someone to a wall; if you were using all of the supplements, then there was a specific feat chain for that maneuver, which meant you couldn't do it unless you planned it in advance and sacrificed your basic competency in order to pursue that feat.
So yes, adding more options does equate to the game taking away options. It's the exact same reason why I would rather play a game that had twelve skills, instead of a game with 400 skills.