You're thinking of a Bard. A barbarian is someone who charges into combat with an axe.
This is D&D. There is no mincing of stereotypes. A barbarian in D&D is a very specific type of character. They rage, and they charge into combat. (Fighters and rogues are less specific types of characters.)
Your stats do not change with age, in this model, so an old barbarian would have needed to survive that long as a barbarian with low Strength in order to reach level 10. Moreover, rogues have several class features that allow them to use Dexterity in place of Strength, and many class features which give them alternative options to fighting; and the lore supports them approaching a problem with in an indirect manner.
If your idea for a cunning "barbarian" is that it cleverly outwits an opponent rather than engaging them in direct combat, then the correct model for how to represent that character within the language of the system is for it to be a member of the rogue class. You don't get better at raging and swinging heavy weapons by cleverly outwitting your opponents.
1) As far as I know, there is no limitation on what you HAVE to be in D&D, hence there being evil paladins, weak but dexterous fighters, Intelligent Sorcerers, and wise Wizards. An archetype does not by any means limit what you are allowed to do, and I would happily allow anyone to play a charming Barb, and am actually bored, just by the idea of another big stupid swinger of great axes.
2) Your stats do not change with age, but there is no reason a player can't make their Strength a 6, and say the reason is that they are old. They could just as easily make a tiny child barbarian with a 6 strength, or a cursed middle aged barb with 6 STR.
3) Have you never read a story that had a crafty viking? maybe not as powerful with an axe as his brethren, but a good enough grasp of people that he could keep up in different ways? That aside, who's to say the old man doesn't plan on taking some rogue training? He has gone Barb for 10 levels, trying hard to fight through his limitations as a late starter on his Tribe's teachings, maybe he finally gives up and looks for better ways to apply his capabilities.
As I understand it, D&D is not a game about "You are this, you must do this". It is a game of meeting challenges in whatever form they come, be they traps, enemies, or poor ability scores. Maybe it is just my point of view, but I cannot see such an enticing character as I have made here as a failure.