Me, I favour the way DA:O did it. It gave you a place and grounded the character in the world without impairing your own characterisation of them. That's the balance I'd like to see more games strike and developers explore.
Yet not a single one of these characters was an origin a'la Larian creations. They are just templates for you to pick from if you can't be arsed to come up with your own.
This sums it up pretty well for me. I would prefer a mix between character templates, for those who just want to get into the game, and possibly backstories like DA:O. Character templates was also found in the early Fallout games and Wasteland use them as well. As for the DA:O backstories they were also acting as tutorials and gave you more of a connection to your character then if you would've started your game in Ostagar.
In relation to what I've stated above, I could potentially love origin characters, but it would require a bit of work. Take Shadowheart as an example. Imagine starting her playthrough soon after her "amnesia" (trying to avoid using spoiler tags), and instead starting the tutorial in a pre-Nautiloid setting. The time on the Nautiloid will still be spent in her pod, in the shape of a cut-scene, and then you get to control her again at the beach. Wyll could have a similar prologue, but with strong focus on his background specifics and how he struggles with the choices of the pact. Lae'zel could be as is but instead jump down on a randomly generated character instead and chose to trust it or to kick it off the Nautiloid.
My point being: If I am introduced to interesting characters and if I'm also given insight in their backstory and motivations, then I can definitely seeing myself enjoying it plenty. Preferably I would also be able to make at least minor adjustments to their kit and stat points, with the exception of Wyll and with certain RP restrictions considering their racial background and backstory.
