So I have played through most of the game as a Dwarf rogue and human warrior. Just started a human mage. And I have to say it just feels better. It is like the only backround where you have a horse in the race. You were a rebal mage. Not just some random shuck that happend to be at the conclave.
I agree.
Non-templar/mage/chantry should have had no business at the Conclave. Dwarves/qunari/elves backgrounds feel out of place somehow. I've heard that devs originally planned to make this game with human only protagonist again, but changed it later. So they've made their general background as spies at the Conclave, but for me that only meant that they are less involved in this whole mess, they are outsiders. I like to get myself involved in the story, so these backgrounds are not for me.
As for human rogue/warrior backgrounds, they're weird as well. You are not of the Chantry, yet you were sent to help some relatives at the Conclave. Do we hear some mention about dead relatives from the protagonist or other ppl later? No. Also, help in what capacity? As a youngest son\daughter of Trevelyans you should be serving the Chantry as is this family's tradition, yet you do not for whatever reason. For me this all means that warrior\rogue is an uninvolved outsider as well.
Human mage on the other hand is very much involved, he\she has strong opinions about templars and mages by getting it first hand, you were legally sent to the Conclave by your Circle and it is very satisfying to achieve what the Conclave was meant to do - resolve mage\templar conflict once and for all. Also, you get more comments about your background than when you're playing as non-mage classes.