You can bet the Arishok of the Qunari or a Dalish clans Keeper aren't traipsing about in a field gathering flowers or grabbing a pickaxe to rush to gather ore they happen to come across. They have people to do that for them. Why would the leader of the Inquisition be much different? Maybe early on, when things are still rocky and being built up, but mid game? Later in the game?
It made sense in DA:O. You were on your own and not commanding an army, didn't really have an army until the final battle. You had to make your own potions and poisons, what not, so you gathered what you needed. It makes less sense in the scope of DA:I, especially when you are a figure of great religious importance. Even if you don't embrace being the Prophet of Andraste, it does not change the fact that you are the head of the Inquisition, the leader, the boss.
Which makes fetch quests all the more annoying too. I had the same beef with games like Skyrim. I'm the Dragonborn. Maybe I'm ALSO the archmage of the college, or I am running the thieves guild or the assassins guild. Whatever, people recognize it, use the titles, and then ask me to go fetch some crap for them.