So you could do a solution like MrFuji suggested, to add specific new quests to the custom character as well, but in the end of the day that would just turn custom characters into a nameless Origin character with a fixed background / in-world interest, [...]
That's a good point. It basically just turns Tav into an Origin character, but probably with a less fleshed out story because Larian'd have to code in a different quest/backstory for each [background] chosen by the player. At that point, why not just play an actual Origin Character...? And it'd mean that you couldn't play a blank-slate character.
Because right now, while it's kind of invisible to players, you actually have to make the characters twice with current system. One with their behavior, quest progression/journal entries, and dialogue as a companion - The other for the same as an Origin (AVATAR). This solution means only one iteration for companions. Or if merged as in last paragraph above, equal amount of work, but not any more.
Even as companions, they could still have tadpoles and be just as they are now. A while back there was a thread about how it feels like the party is missing a leader, and questioning why the companions would even want to follow Tav, or how they have every reason not to but for some reason seem to follow them anyway. An MSQ could help underline Tav as the group leader and mitigate some of that party synergy dissonance people mentioned in that thread as well, potentially.
All fair. It would be ~zero work for current Origin companions, and less work for any future companions Larian plans to add. And yes it would be one way to "underline Tav as the group leader." Though that could also be done through additional dialogue where the party, somehow, nominates you as a leader. Maybe you're particularly convincing (Charisma rolls) or the party has conflicting interests (SH and Lae'zel) and thus make you the tie-breaker.