A dex barbarian is already "allowed", it just isn't optimized.
Building a barbarian without Strength has an immediate impact on the way Rages work, since you lose out on the bonus damage (+2 per attack, growing to +4 by level 16) and on Reckless Attack (2) (which is itself a bit of a wash anyways, with the benefits being countered by a penalty). That’s the cost of doing business. Since Dexterity attacks synergize with Unarmored Defense, this is a choice that is still made, and there is nothing especially unconventional about a Dex-based barbarian.