Leliana wants everybody in the world to convert, and says as much. So... that's always been the official position of the Andrastian Chantry, but the thought of what her position could lead to makes me queasy. Not everybody wants what the Chantry is selling, and Leliana doesn't seem to understand that. She's also way, way more cynical about her faith than she used to be and doesn't have a problem with offing people who threaten the institutions she protects, or flat out lying to defend her secrets, which puts her version of the Chantry about one bad day away from piking the heads of heretics and denying reality to serve its own ends. Sunshine and rainbows make great cover for zealotry. She also expresses a strong desire to become the Divine, which scares me. Nobody should want that kind of power. I'd rather see her quietly retire and take up nug breeding as a hobby.
Cassandra is willing to accept that some folks just aren't interested in her faith. I was way more comfortable putting her in the driver's seat, knowing that she would work on putting her own house in order and not try to play "let's go convert all the heathens!" She has a lot of firsthand experience with zealots who got far too wrapped up in their faith to do what was right, and I think she stands a better chance than Leliana of not falling into that trap. She also comes from a formal position within the Chantry heirarchy- after Lucius bites it, she's effectively the senior surviving Seeker, and instead of clamping down she actually starts airing her order's dirty laundry as soon as she takes possession of the Big Book O'Secrets. That was pretty encouraging.
Vivienne... I think she'd be the most technically competent as Divine (being an administrator already in her position as First Enchanter), however I just couldn't convince myself that the Chantry would accept a mage as its leader. Yeah, the game lets you do it, but it goes against all of the established lore. She's also got this disturbing little "I know better than everyone else about everything" bit going on. Plus... I don't know about anyone else, but when I talked to Vivienne about who should be the next Divine, she was pretty dead set on putting Cassandra in the Big Chair. Which made it doubly weird when the "support Vivienne as next Divine" operation then appeared on my war table, because nobody including Viv herself had even mentioned the possibility.
So... yeah. Cassandra FTW. She's got insider knowledge and strong faith coupled with a healthy sense of restraint and at least a willingness to understand other beliefs before she dismisses them and the people who hold them. She'll be the most willing to take good advice from her staff, but at the same time be willing to knock her advisors' skulls together when they go too far. IMO she's the best and safest choice that the rest of the world outside the Chantry can also coexist peacefully with.