I get it, you don't want to ruin the experience of "earning" your own loot. But when you get an exotic or legendary item that is not for your class, could you at least give the option to trade it among the people you play the strike with? Specially among Clan Members.
And I believe it'd be very healthy and a valuable for anybody to be able to trade among clan members. Make the trading a balanced tool where you can only trade item of the same value. You say everybody would gear themselves in just a few hours? Then limit the trade to 20 common items (weapons parts, consumables) , 10 green, 5 blue, 1 legendary gear & 1 weapon each week. And leave exotic for only trade within the strike party members.
What I ask is not far fetched because I've seen this working within some MMO's. You wan to improve the idea of social cooperation? Then other than shooting with more dudes.... this could be a thing too :)
There, my two scents.
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If they installed a trade system it should only be possible through the vendors as long as you have fully upgraded your weapons or gear.