Let's be honest here. For the most part no one uses exotic weapons primaries outside of pvp for anything that doesn't requires a quick shield take down or a burn for a certain element. I myself only use them in PVP and the Strike playlists.
Sadly, when you get into the later stages of Destiny, this winds up making a lot of exotic primaries useless. For example, whenever I got into a solar burn nightfall, I am not looking at my bad juju, Monte Carlo, or Hawkmoon, I go straight for either my abyss defiant or VoC.
Wouldn't these exotics be more sought after if they actually offered an element to go with them?
I know the argument against this is "But then Raid legendaries would become useless" and that isn't necessarily true. Some people prefer one type of a gun to another, and even then there are different builds within in each class of weaponry and people have different preferences there as well.
"But what about Raid Exotics?" They still have their niche and place as well, as it is the only primary fusion rifle you can get is the Vex Mythoclast which makes it more unique than others. Nechrochasm is the only Auto-rifle in the game that gets a firefly perk at all as well.
Neither the raid legendaries or exotics would lost their place among the other weapons just because everyone has a preference to what weapons they wish to use.
Edit: Great to see a lot of positive feed back on here. Seems everyone is in agreement (for the majority) that every exotic primary should have an element to them.
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Exotics should deal more damage than legendaries. If a weapon has an element associated with its damage it should do less damage than an exotic with pure kinetic damage. How you obtain an exotic is a different matter that is much complicated tha Xurn v Earn debate!