I think that the main benefit of Barbarian/Rogue is having Advantage on Str checks from Rage, and double proficiency bonus on Athletics from Expertise. Plus possibly higher AC from prioritizing Dex/Con with Unarmored Defense.

The DPR for straight Rogue, straight Barbarian (using a two handed weapon), and multiclass Rogue/Barbarian is going to be roughly comparable, once you account for Feats, Sneak Attack Finesse limitation, etc.

Honestly, I’d be fine with a Beast Barbarian or Rogue/Barbarian using Shortsword/Claw attacking with each twice per round (which legitimately gives a big damage boost at low levels) without the silliness of dropping a weapon every turn. Its limited by Rage uses. Claw is a Str based weapon, which makes Dex synergy harder. It uses the bonus action. And both classes are otherwise outshined by most other builds at high levels.