Okay, firstly, the quest you're mentioning in Dragon Age Inquisition you're way off. The quest is called Shallow Breaths and was actually that the guys kid had joined the cult, which believed that the Maker had forsaken the world and were attempting to regain his favor with praise or somesuch, and the guy's wife was having a relapse with her breathing disorder. What he asked you to do was to find him and get the potion he used to make for his mother.
When you finally track the kid down he explains that he didn't think it would be a problem because his mother had not had that issue in a very long time. He then gives you not only the potion and the recipe for it. You then give the potion to his father SO HIS WIFE DOESN'T DIE.
And that's just one quest.
There were dozens of side quests that were very detailed and had back stories to them. I can think of two that were very similar to quests in Dragon Age: Origins.
The Ballad of Lord Woolsley and The Mercenary Fortress/Deep Trouble.
In The Ballad of Lord Woolsley a man in Redcliffe asks you to find his ram. Pretty easy to do because it's very, very differently colored from other rams. Also, he says that the Ram has given his family advice for generations. It sounds like he's crazy, but then you go find the ram. Now, the game doesn't tell you this, but you actually have two options here. You can just walk near the ram and tell it to go home and it does. Or, you can attack the ram. If you attack the ram, it transforms into a Rage Demon. After killing Lord Woolsley, the quest ends saying the owner 'must never know'. This is an example of a quest with a very subtle extra story that made Dragon Age Origins a classic.
The Mercenary Fortress/Deep Trouble are two separate quests that are literally tied together. You wander along the road and a scout tells you to watch out for some very odd bandits. You fight your way through them and find orders from their boss and go to their 'fortress'. You find out along the way that the bandits were actually mercenaries hired by a Dwarven Carta to keep refugees and your own forces away from their Red Lyrium mining operation. Thus begins Deep Trouble, where you find some old Dwarven Ruins and a nice little Carta that you have to fight through. You find out that they've been monitoring you as well as mining red lyrium all over the place. You also have your (potentially) first encounter with the darkspawn in the game, and a vault full of goodies.
Seriously, there are dozens of quests as good if not better than these two, but the thing is you have to actually EXPLORE THE WORLD to find them. Just like in Dragon Age Origins, they don't throw themselves at you. YOU DO NOT SEE THE DEPTH OF THE GAME IF YOU STAY IN THE KIDDIE POOL PART OF THE GAME.