So we have six primals and six elements - one for each.

Our existing primal weapons have a chance of dealing additional damage of their specific element.

Ramp that feature up. Make it so that all the damage dealt by a primal weapon is of its specific element rather than slashing/piercing/projectile/blunt, while allowing the mage weapons to capitalize on the traits of their specific element if within that discipline or other unique benefits if outside the discipline.

In short, make the weapons situational and useful in exploiting elemental weaknesses - including against rival primals.
A Shiva weapon would then work wonders against Garuda, who has an ice weakness. Likewise an Ifrit weapon would show Shiva who's boss.