Personally I would overhaul the War Table and Power and Influence mechanic to merge them together. Instead of abstract power points that we get for doing just about anything, divide it into three categories. Forces, Secrets and Diplomacy.
Say we have a meter that measures the strength of each section, and the total represents our influence. The numbers on the meter would represent how many soldiers, spies and diplomats we have. We can boost these numbers through the various agents we recruit (instead of time reduction). Let's take this hypothetical scenario with the Hinterlands:
Get Vale's irregulars (+25 to Forces)
Get Speaker Anais and her cultists (+10 to Secrets or +10 to Diplomacy depending on choices)
Get Master Dennet (+5 to Forces)
Etc.
I put a high number on Vale's Irregulars because of the amount of work required to get them. All three numbers added up would determine our influence and unlock the story missions.
Next thing: instead of an advisor being locked to one mission for a given amount of time (feels artificial), the various War Table missions require an investment of points from whichever advisor we are dispatching. For example:
Deal with the Angry Varghest: Send scouts (-15 to Secrets) or send Soldiers (-15 to Forces) (no diplomacy option). Now, this subtraction is potentially temporary. In this particular mission, if we send scouts, the agents die but the threat is dealt with. If we send soldiers, the threat is dealt with with no losses according to the report. If scouts are picked in this case, the score subtraction is permanent (because, hey, they're dead). If soldiers are picked, we get our 15 points back (great success!). So we could send spies, soldiers, etc. on more than one mission at a time. (hello, resource collections)
Such a mechanic would allow us to feel the weight of our decisions as well as give us a clear visual of which of the three branches we favor. Imagine seeing your Forces score go way down if you got the Wardens killed in a War Table mission (-100. Ouch!) Instead of the influence reward we get from some missions, we would get a boost to one of our three departments.
Now, of course this would have the potential for min maxing (which is ok), but let's say on a first play through, you have a bunch of screw ups and you get almost all of your spies and soldiers killed. What do you do? Is it game over? Well no. You could get allies and boost your Diplomacy score. How? Well, the Dwarves of Orzammar would be willing to send aid if you helped them recover some artifacts for them (mosaics collection); A Nevarran Mortalitasi is curious about that strange temple in the Forbidden Oasis (shards collection); An Orlesian noble would pledge some of his forces to you if you cleared his land of rifts (self explanatory).
That way, all those pointless fetch quests could be tied into the plot as a way out of a hole if you screwed up at the War Table. You could still do them to boost your Diplomacy score anyway (min maxing), but they would remain largely optional.
Maybe throw in a cutscene or two with the advisors after a major loss or gain in their respective score, perhaps even some grumbling if one department is heavily favored, and that would add a good deal of flavor.
Enough rambling. Just ideas I'm toying with.