But that's just it, the "other things" shouldn't be the stupid crap that every grunt and scout are more or less capable of doing.
The Inquisitor is a capable Mage / Fighter, so it stands to reason that he will help when he can, but there are priorities and tasks appropriate to the individual. You don't send jet fighters and attack helicopters to give speeding tickets, and you don't send a general to clean the latrines.
It makes sense for a wandering monster slayer to take monster contracts and the occasional weird side quest.
Hell, even the card game makes a certain amount of sense as an activity to wind down and relax from time to time.
But a supposed general gathering all those odds and ends is just a little bit difficult to take seriously.
I don't really understand why the protagonist had to become the "chosen one" and a leader so quickly, or at all, but at least be consistent.
That's why I think that the War Table could have worked if it was a more in depth system on the scale of XCOM's overworld.
Imagine being able to control troop assignments in each region. We start by setting up a base of operations (much like we did in the base game) and dealing with an initial large event (mage/templar battle, large rift portal, dragon, etc.). From the war table we could then manage our limited forces (we wouldn't just magically have all the forces we need for the task at hand) by sending them to do the kinds of grunt work quests that we did in DA:I as well as other generic stuff like repairing damage and protecting villages from conflict, gradually expanding our influence of each area. Essentially, the gameplay would work like a simplified 4X game, but instead of directly controlling any troops, you just assign numbers to outposts and occasionally complete the same old adviser missions.
If we wanted to get more advanced, we could have different strategy layers for spying and diplomacy as well, which would involve keeping the existing governments of an area happy during its inquisition occupation. Or we could just go with a more simple "diplomatic favor/blackmail tokens," which could complete a single War Table action for free. Or maybe something else.
Either way, our forces might periodically run into a problem like "accidentally waking the mummy's curse," and we could go in and deal with it (or not) by completing a cool sidequest. Of course, If we want to keep exploration a high priority, then these other quests could be discoverable via our own exploration without the need for the War Table to alert us.
I can understand doing crap quests in the Hinterlands when we have practically no one working for us, but once we got a foothold, the gameplay should have changed with us.
We'd need to get into specific cases. Most of the cases I remember are not things I diverted to complete. You might have to divert to turn them in, but you'd do that after finishing the local mission.
I seem to remember the opposite. Aren't there a couple areas that don't even have a "local mission?" I'd be inclined to agree with you if I didn't strongly remember getting bored with the quest structure. Had most of these missions truly been on the way, I don't think I'd feel that way. I could be wrong though. Like you said, we'd need to get into specifics in order to sort my recollection out.
Even so, the tasks we had to complete, on the way or not, could have been more interesting. Though I decry Witcher 3's repetitive quest structure, I found it far more bearable because each sidequest had a good narrative hook and generally something fairly unique to it.