I think roaming around the Conclave for 20-30 minutes would be nice (the horrible tutorial level could be cut shorter as well). You could talk to people connected to your background (friends/allies who came along) to establish your character's views and beliefs early on, maybe even take a look at the Divine so you would actually give a damn about her murder (same goes for the other victims).
In general, DAI worked okay for me, but that's because I had my character's backstory planned out weeks before the game came out. I still prefer the way Origins did it, it helped me get immersed in the world more. The Herald in DAI did feel more like a decoration piece, as someone here said, like s/he didn't even exist before the Conclave.
I had an idea about that a while back, when I was initially disappointed with the backgrounds.
The conclave would have been a playable stage before the explosion, and your race/background would change the initial events and how you found yourself to be in the chamber with Corypheus and the Divine.
Human Warrior/Rogue. You are at the conclave representing Ostwick nobility. You get to greet the divine before entering a gathering discussing the Champion of Kirkwall. It is mentioned that a "Serah Tethras" is being brought to the conclave to testify (and autograph)
Human Mage. You are at the conclave representing the circles of magi. You are introduced to the stage by Seeker Cassandra who leaves your party because she has to fetch "a loud mouth dwarf" back in haven. You get to explore a bit before the same meeting.
Elf. You are sneaking around the conclave. Outside you run into Solas who reveals that he noticed your presence a while ago. He tells you that he will be leaving shortly. You observe the meetings from above, Batman style.
Dwarf. Same as elf, but you are found out by Leliana instead. Trying to escape her you stumple into some forgotten hallways where you discover red lyrium under the temple. You lead her away from the conclave to escape her, some of your fellow Carta create a diversion that lead her away from the conclave.
Qunari. You are a part of the mercenary guard. You are not permitted to enter the conclave and are introduced to the level by Commander Cullen who then leaves to see to his own forces at the forward camp further down the hills. You decide that guard duty is boring and try to sneak your way to the meeting.
The meeting is then interrupted by Grey Warden mages that warn people that they have been alerted to Darkspawn presence in the temple. Panic erupts and you try to escape along another path. On the way you hear the divine protest to her captivity and decide to see what's up. You enter a room and it all ends in a flash of green light.