Actually, I believe you heavily overestimate the likelihood of non-human races, specifically in Mass Effect. I'll explain:
Perhaps you haven't played SWTOR, but there is a reason that every race in the game is humanoid and all can speak basic. It's deliberate and part of BioWare's design. You'll never be able to play as a darkspawn, elcor, hanar, volus, vorcha, demon, or anything of the sort as a main protagonist. Why? Limitations. The reason why many perceive BioWare is so "pro-human" is because humans actually have a lot more variety and flexibility than most races even though many argue humans are "stale" and "generic."
Whether we look at diversity in looks, a wide range of animations, the ability to have romances, speaking the common tongue, there are many reasons why humans are preferred over anything else. This is why there will never be a playable jawa, ewok, wookiee, or anything in SWTOR that cannot fit the basic standards BioWare needs of a protagonist in the story. The same applies to Mass Effect, which is why it's unlikely you'll ever see multi-race. With DA, BioWare can get away with it a bit more as qunari/elf/dwarf are just fantasy humans. Even with that cosmetic distinction, it has been shown that race rarely matters and overall has little impact or influence on the story.
As others have stated, if race doesn't matter and only adversely affects the story because BioWare cripples the voice acting and dialogue, I don't see how it's a benefit to anyone except those who will headcanon regardless. If you want to fabricate your own story and completely contradict what BioWare games do, that's your prerogative. However, I purchase BioWare games because I want to see their stories unfold with amazing characters. Multi-race merely is counterproductive and hurts why many love BioWare games to start.
Well you could see options like Turian, Quarian, Salarian Asari, and Drell.
As far as animation goes, they can re-use a lot of the animations if the other race is humanoid which is why the playable ones generally are. Mass Effect 3 MP offered playable Turian, Quarian, Human, Salarian, Asari, Drell, Krogan, Batarian, Vorcha, Geth, Volus, and Collector. While not all would work in the dialogue heavy SP, a number of them most certainly still would and you could keep the same voice actor for it.
A lot of things don't matter and have little impact on the story. I recall a conversation in which Cassandra asked my Inquisitor if she believed in the Maker. At no point in the future was that ever referenced again. Should we also remove all flavour dialogue because they don't have an impact on the story? They're nice little RP pieces, which even if it's story driven and cinematic Dragon Age is still a RPG.
Hell, a lot of the actual choices you make in quests don't even matter 5 minutes after you're done that quest. Most of the choices you make in Mass Effect get acknowledged in an email in Mass Effect 2 and that's it. They just become flavour text.
As I said before I don't see how multi-race negatively impacts the game considering Origins/Inquisition vs Mass Effect. In my opinion, the Inquisitor's voice acting is not worse than Shep's or Hawke's, and the dialogue in all games has its ups and downs.