Primary weapons that are below exotic-tier do not need an elemental damage modifier.
There is no substantial reason to have elemental damage on a primary weapon outside of fitting the weapon into a tied theme, e.g; Zhalo Supercell dealing Arc damage due to it being an electricity-themed weapon.
[edit:]
My standpoint right now is that elemental damage in Destiny has little purpose beyond dealing extra damage to enemy shields. Not shields in general, but shields of a specific type to be specific. Right out the gate the usefulness of the elemental damage is limited.
Look at it in a broad perspective, and elemental damage in Destiny is small, inconsequential, limited and simply boring.
It does not add anything to a weapon. It does not give it a personality. It does not make the weapon standout more. It does nothing.
Give elemental damage some actual use and nature and it could be worthwhile having on a primary weapon.
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*cough* raid weapons *cough*