Some quests steps (e.g: Forsaken expansion's Ace of Spades quest involves killing 5 invading players in Gambit and 25 precision kills in Crucible) require players to participate in PvP activities that they would rather not engage in. Would it be possible to simply add a PvE option to any quest step that initially requires PvP? This would not take anything away from PvP players & would allow PvE-only players to complete a quest step that they would originally shun.
I think that the quest for a weapon should be representative of where you'd most likely be using it. The ace of spades is a very versatile weapon which works in pretty much any environment, PVE, Crucible, and Gambit alike. It makes sense that to get it, you have to participate in all of those actions. Plus, for that particular part of the quest you only need 25 precision hand cannon kills (even if you're bad at PVP I can't see this taking longer than a couple hours, and possibly in one game if you're good) and in gambit you have the option to get PVE enemy kills towards the quest OR invaders. When you look at quests for weapons like Thorn, Last word, Chaperone ETC, most of the quest is in PVP. Why? They're mostly used in that environment.