While I agree about the Witcher games pulling it off extremely well, I don't think a voiced player character makes fixed race a necessity. It does, as you said, make some personality specifics a necessity, but even with the Witcher example there were more than enough choices in how Grealt could act in a given situation. If they wanted to, Bioware could easily restore race selection and still have a voiced PC (same male voice, same female voice, regardless of race) with enough variety for each race and without rewriting entire dialogues to account for each race.
The problem is, I think, that Bioware's design team is so stuck in that cinematic trend that they have completely forgotten there are other ways to add variety to different races, like introducing racial traits, bonuses or weaknesses and/or special abilities available to each class, depending on the chosen race. Think something along the lines of VTM: Bloodlines and the various abilities each clan had - like the Toreadors having a seduction bonus, the Ventrue having that Mind Control dialogue option and in general Mind Control combat abilities, the Nosferatu being great hackers and lockpicks, but forced to use the sewers to move about and the Malkavians being completely insane. I'm sure something similar could be added to Dragon Age, only perhaps less drastic than the above examples. Combine that with some minor dialogue variety, a la DA: Origins, and the result would be pretty neat.