Barbarians come from uncivilized areas, not inside of them. Let's look closer at your street urchin and the barbarian class. These abilities are nonsensical for such a person:unarmored defense, fast movement, brutal critical, indomitable might, and primal champion. That's the problem with re-fluffing classes into something completely different. There are usually abilities that don't fit.
Just to confirm, you're using "Barbarians come from uncivilized areas" as your definition of which Barbarian class fluff is essential and which Barbarian class fluff can be ignored without special permission? Do you have a particular citation for that? It's definitely not listed in the PHB as being more important than any of the other class fluff, nor does it closely match any of the definitions of the word "barbarian" I'm seeing (mostly historical earth definitions, implying a current state of being uncivilized, or implying a relative judgement on the value of other cultures.) How are your players supposed to know that's the definition you're using in order to know that at your table, moving to and living in city is fine for a barbarian, but they can't have been born there?
And I see absolutely nothing inappropriate for a feral street urchin about any of the Barbarian features you listed. Surviving childhood on one's own in the underbelly of a major metropolis could easily be a harsher and more-violent existance than being raised by an isolated tribe or joning a group of dwarven berserkers.