I don't mind having companions with interesting and personal stories - but they need to be strongly secondary to the core story of the game, and the core story of the game needs to focus on, and make the player character feel like the actual protagonist.
The 'origins' aren't companions and don't feel like them, in that sense; it's very obvious that they are all designed to BE the protagonist, far more so than the player character, and the result is that they feel like mary-sues, who are every so much more important than anything else that's going on. I'm happy to get involved with their personal stories and side quests, but they should be that - side quests that are complimentary and related, but secondary to the main game's story, which the player, is ultimately the lead figure in; companions follow the group's leader (the player), *Because* that focus is ultimately more important than their personal story, and they're aware of it; it doesn't detract from the importance of their personal goals, but if their personal stories are so much grander and more important and focused upon than the core, then the reason for them follow our lead evaporates, and that is what happens here. When these characters travel with us, the extreme majority of their interactions revolve around demonstrating through dialogue just how much superior to us they are, and we rarely, if ever, get any opportunity to do anything beyond simply letting them; I'm int he process of testing the new interactions now, but the overall mode of their presence and cooler-than-you importance hasn't changed.
When you say templates, this brings to mind what I was thinking of and hoping for when they said 'origin avatars': What I'd really like would be if we could, rather than playing AS the origin characters, select from optional origin templates when making a custom character - and yes, they might load up when selected with a default name (Gale), and a default race and appearance (human male), but would be entirely renamable, and recustomisable to the character we'd like to play, and we could slip into the role of someone from that origin, but as the person we want to play as. The game should ultimately be about our characters - not Larian's.
If we don't pick an origin template, then we should still have an interwove story and background (or one of several to pick from), and the game can certainly predefined *some* elements of it for us to help work our custom character into the setting more, while letting us define and establish other less integral aspects of our history as we go - NWN2 did this to great effect, as an example. It would be the 'default' origin story, and wouldn't be represented by a companion if it wasn't taken.
As for the other suggestions here, it's an interesting idea, but I don't see how they'd be able to rectify that many different angles and motivations to align into a coherent story without functionally having to make multiple entirely separate games - not that it's not interesting (it is!), I just don't see how alternates that are that different could all channel into the same base game and feel believable.
In older editions there was a specific type of mindflayer that, after ceremorphosis actually came out the other side visually unchanged and completely resembling the person they were before-hand, and even to a certain extent retaining some of the sense of self and personality of that person. They could act as infiltrators and the like, and passed entirely well as the person they used to be, but still had to feed on brains and psychic energy. They were also prone to breaking away from the goals and plans of their elder brain and going rogue... Hmmm...