I know I am digging an old grave (how cheerful I am today) but I am at peace with Cullen's portrayal in DAi. Before the game was released it was hard for me to imagine him voluntarily leaving Templars, the order was deeply ingrained in him and he would have to go through a major identity crisys if he severed the ties. It must have happened gradually, over the course of few years after Hawke's shenaningans, when he was left in Kirkwall's ruin of a city, unfortunately off screen, which is a pity but that's how the writers rolled. *shrug* Maybe over that time he discovered that he enjoyed that role more, being in command, taking responsibility for others without the order, the rules and the whole baggage. He joined Templars when he was a very young boy and those dramatic circumstances might make him aware that there was an alternative.
As for whether he'd be a good candidate to restore the order? I am not sure anymore. I loved this part of his character and when I imagined him in the game (admittedly I though it would be just an extended cameo, as in da2) I could see him being involved in making a change, cause he was there and saw all the wrongs. However, in his almost 20 years of being a Templars he really went through some serious stuff. Considering that we have also witnessed a change in his attitudes towards the order, him slowly becoming disillusioned and simply disappointed how the ideals did not match the reality, it is not out of character that he would get burned out and lacked determination and idealism to push the order in a new direction. He was dutiful and dedicated in da2, 100% invested in the order that it felt there was no place for off duty Cullen in there.
In dai on the other hand it seems like with his decision of leaving Templars he gained a new energy and enthusiasm for life. He is still haunted by his past (nightmares, regrets, what ifs etc), but he is seizing the moment, trying to salvage some happiness (with Quizzy or not) and he's not letting anything (even his addiction) to overshadow it. Lyrium is a big problem here, since you cannot be a Templar when you try to stay clean of the stuff. I don't find the seekers' alternative that much better either. Initially I was slightly taken aback how considerably pain free the whole decision about the lyrium was in game (especially after all the grimy stuff we have learned in two earlier installments), but I guess devs did not want to focus on this particular aspect. I can headcanon that Cullen is extremely strong willed, driven and still quite young, hence capable of managing it to a better degree than some.
Tl;dr In character or not I just love to see him happy, regardless of where his future lies. It is quite a change to see him relaxed and smiling that I don't find his his portrayal overly grating. Curious to feel that I care so much about future of a bunch of pixels but here you go.
And since I might not survive till 4k celebration (got up at 5am today) here's my Quizzy being all happy romantic with Cullen. She absolutely adores him.