I would like those choices to have an impact on the things that the player is required to do to solve problems, with some paths only being open to certain origins. But in the end, the result would be the same: the problem is solved and the player moves on to the next part of the plot. If we use DAI as an example, perhaps the Fade sequence could have played out differently if the PC was a mage, with some other magey elements (like possession) adding to the danger, or something along those lines. Perhaps in the Mark of the Assassin DLC the sneaky bits would only be open to a rogue player, and so on.
They did this to a very limited extent in Champions of the Just. A mage PC has the option of recognizing the demon right off the bat and straight up refusing to respond to the confrontations Envy forces as you progress, thus giving the demon nothing to hang their mimicry on. Non-mage PCs have to choose an actual response, which would presumably make it easier for Envy to replace them and make it easier to fail the quest. Not almost impossible to succeed, but definitely easier to fail, which is probably the direction that the devs should err on if they're creating distinctions between race/personality interactions. Shortcuts for specific characters can work, too, if they're not overly generous with the reward, like the Omega DLC for Mass Effect 3, where the Engineer Shepard was able to cut through a hacking scene immediately with a sarcastic comment, while other classes would have to face repeated temptation to sacrifice innocents rather than take the time to complete the hack.
I think they've just been really hit and miss. Some stuff makes sense and some doesn't, and it's sometimes hard to tell if it was intentional for gameplay reasons, if it was overlooked, or if there was no way to reliably implement it due to resource constraints. It made sense that a mage PC would be able to refuse to even speak with a demon, but a non-mage PC could be goaded into responding. Limiting learning about the true meaning behind the vallaslin to elf Inquisitors made sense; it didn't make sense to gate it behind the Solas romance, as opposed to a high-approval Solas. It didn't make sense to have dwarf Inquisitors act like dreaming was something they've done before, I don't care how important it was that the first Skyhold conversation with Solas was supposed to happen in the Fade.
When they do it right it's pretty enjoyable, and fun to experience, especially on a subsequent playthrough. I wish they were more consistent with it, though.