Yes I do.
You are the Inquisitor period. You are limited in how said inquisitor can behave to a certain extent which is why i am surprised you like the game because the limits on the character are the same as most Bioware games that you complain about. As with ALL bioware games you have a very predefined character that as always can have limited variations on their background. It is impossible to play a Qunari who is an ex member of the Qun. You were NEVER part of the Qun your parents were Tal-Vashoth. You can not be a Human commoner that isn't an option you are a Noble period no ands ifs or buts about it. You can not play a city elf that isn't an option. The point i am making is that the divergent backgrounds for character concepts just don't exist in DA:I. I personally don't feel this is a problem because I have NEVER ascribed role playing to player agency or character creation, especially character concept creation. They key has always been taking a role other than your own and making choices based on what you think would motive this new role.
You are limited in player agency in DA:I with the choices you can choose as well. In DA:O it was possible to be a psychotic warden. It isn't possible in DA:I to be a Psychotic Inquisitor again clear limits on player agency and yet again I think this is a sign of good game development. The Narrative in DA:I has no room for a psychotic and thus the player isn't given the option which I think is a sign of strength of design because they are not trying to force in agency for the player where is doesn't fit the story they are trying to tell. This limit is part of how predefined the Inquisitor is, I found the inquistor as predefined as Shepard, Reven in Kotor and so on. They have all been relatively predefined with variations allowed within limits.
It has always been my contention that the player has a responsibility to make a character that FITS the game not just make a character they want to make because they think it is a cool concept. Players are minor partners to the cRPG story and it has always been that way but now with more complex and detailed stories being told by games it is more difficult if not impossible to hide the fact the player doesn't really have as much agency as they thought they did.
I try to make characters I can play, which means I look for the gaps that are available. I don't intentionally create unsupported characters.
My complaint with ME and DA2 isn't that there are no gaps (though there aren't enough in ME3); it's that the gaps are invisible until it's too late to take advantage of them.
I do make characters who fit the game (if possible). What I don't do is make characters who fit the narrative, because I don't grant the narrative any particular importance. I won't even necessarily concede that it's there; instead, I'll see a great many discrete events which might not be related.
Yes, at some point in DAI my character will become the Inquisitor (assuming she lives that long). But that tells me nothing about whether she wants to be the Inquisitor, or whether she thinks making her the Inquisitor was a good idea, or what she thinks the Inquisitor's job is, or if she cares. Being the Inquisitor might not even be something she considered relevant.
Being the Warden is arguably more of a burden, because being a Warden is ultimately fatal (though that can also create motives).
I'm not claiming I can design any character and see DAI support it, nor do I think CRPGs should even aspire to do that. What I think CRPGs should do is leave us alone to engage with the content as we see fit, and I think DAI (and DAO) do that really well.
At least on the uncharted worlds, so does Mass Effect.
BioWare has made CRPGs that allow us near total freedom to define our characters' backgrounds, but they've only done that twice, in NWN and KotOR. Clearly that's not a make-or-break for me; KotOR is only my fifth or sixth favourite BioWare game.
Oh, and if we thought we had agency, we had agency. No one else can judge that. I made all the decisions for my characters that I thought I did. Remember: Death of Author. The devs' intent never matters, not even a little bit.