Good morning!
It took me a while to it to finally sink in regarding which of your siblings makes it through the prologue -- and why. DA2's writers wanted to make sure your Hawke has a vested interest in the Mage/Templar war.If your Hawke is a mage, there would be no counterbalance to the conflict if Bethany survives. Likewise, if your Hawke is a rogue or warrior and Carver lives there wouldn't be a compelling reason to support the mages.
My canon Hawke playthrough had him as a rogue who supported the mages. He was very protective of Bethany after the events that started DA2. Given a choice of sending her to the Circle or making her a Grey Warden, I chose the Circle. Hawke wouldn't have known that being a Grey Warden tends to shorten your life span but I felt that with the Kirkwall Circle destroyed Hawke's sister has the chance of freedom and the prospect of a long life.
Since it appears that there's not a clear-cut reason who to support in DAI, I'm wondering if your companions will be the ones who will try to influence you. I was hoping the advisers might have a say-so as well, since Cullen looks to have had the most experience in the conflict.
Morning 
I have a hard time not choosing Carver as my go to sibling and thus Mage as my class (not that it takes twisting my arm given that Mage is my favorite DA2 class) because to me he has the best development in the game. He's a little **** for sure, but it seems like he's had to sacrifice a lot in order to keep his family safe and when I befriended him and sent him off to the Wardens he finally got what he wanted even if it wasn't the exact way he wanted to find a purpose.
A little headcanon I have is that either my Warden of Nathaniel took Carver under their wing and showed them that being a little brother wasn't so bad and that sometimes you could live besides your siblings in the limelight. I also think my Warden's more sober attitude was a little refreshing to all the snark my jackass (which is a term of endearment for my Champion) Hawke was often spitting. Just my Cousland, Nathaniel and Carver carving a swath through Darkspawn 
Anyway my mage Hawke was a mage supporter, but while he was very fervent in his ideals at the start he was a bit less certain by the end of the game. His mother had been killed by a mage, he'd potentially learned that the Magisters were indeed the cause of the darkspawn and then the whole thing with Anders made it harder for him to not see that there was a point to the whole Templar thing. That said he believed that all mages shouldn't be judged by the events of the past or the workings of the insane and so he sided with the mages (though he had no particular hate for Templars anymore than anyone else who tried to kill him.) He also let Anders live and felt like he'd betrayed Sebastian which he regrets somewhat.
I think if it's as one sided as it was in DA2 or Origins I'll be siding with the mages again, but if its a bit more complicated when the decision has to be made I might side with the Templars.
I wonder if instead of sending Leliana you send a Templar like Cullen and because of his abilities he's actually able to subdue Alexius?