I treat the "barbarian" part of Barbarian as more of flavor than a literal barbarian. They have abilities that would help them in a primitive culture, but that doesn't mean they are exclusively found there. Where a Bighter relies heavily on training and technique to beat his foes, a Barbarian relies on instinct and superior physical ability. Barbs are just plain faster, tougher and stronger than their opponents.
As for multiclassing into Barbarian, it seems like a natural way to learn combat. Basically, without specific training to be a Fighter or Ranger, a warrior will act like a Barbarian. So by simply wading into combat and surviving, you can probably become a Barb.
Per my view, when I DM for Barbarians, I allow free multiclassing, and may or may not drop the "non-lawful" restriction. I do ignore the illiteracy restriction.