I have a thought: even without demons and Meredith and red lyrium, Cullen was never temperamentally suited to be a templar. Or at least, not one assigned to a Circle. I've been playing around with the Magi origin and it's just now dawning on me what a political place the Circle is. Irving doesn't really want to Tranquil Jowan, but what he really cares about is seeing the Chantry also suffer. Leolach (sp) wants to be seen as competent more than anything. Lots of jockeying for power, between mages and Chantry and among the various fraternities. I get the impression from marketing info that Cullen has little patience for that sort of thing. He might have been happier as a common soldier or Gray Warden.
From my point of view Cullen is someone who wants to protect people. At least this is how I picture him as a young and naive Templar. So, I think that he is very well suited for the job. It's true that there was a lot of political powerplay going in the Ferelden Circle but Cullen wasn't really affected by it. Maybe if he had been promoted at one point...
In Kirkwall he had probably more contact with politics and I believe that he grew into it. I can't imagine that he would be a military advisor if he had no sense for politics at all. I even believe that he could be quite good at it. Cullen isn't Alistair after all who never wanted to be a political figure and always prefered a life on the road. And even Alistair grew with his tasks.
There must be a reason in the past of Cullen. I'm very curious about it.
Perhaps he had no choice.
I hope we find out more about his backstory in DAI. I'm very curious about the reason why or how he joined the Templars, too.