I'm voting for making the custom character more interesting for sure. I wrote some time ago here my suggestions about how exactly:
1. The custom character (CC) importance.
After quite a lot of thinking I realized that it doesn't matter how much you untie the origins from the main plot, it doesn't matter how much you tone down their stories, the problem just is not really in them. We had quite a lot of oh-so-special companions in other RPGs. The only difference is that we couldn't take them as our own character that doesn't really matter in comparing their importance with our main character's importance there. So it's the CC himself lacking his own upsides added in the plot. And here my thoughts on how it can be changed without taking away the freedom to have whatever backstory for our CCs in our heads:
1.1. Add the very special connection to the main plot only for the custom character.
It should be something the character himself couldn’t know before the main plot came into motion. Maybe he was observed by The Absolute before, got favored and now he has a different kind of tadpole that gives him special powers over the rest of tadpole users. Maybe he has some relation with whoever The Absolute is (yes, that’s somewhat BG1 all over again, but still better than being Tav-named nobody).
1.2. Make the custom character the only one type of glue to paste them all together for companions as continuation of those special tadpole powers.
You have Lae’zel and Shadowheart hating each other, Astarion being so paranoid that aggressive at the first meeting, Gale jumping on our character from the teleport, Wyll fighting goblins before becoming the part of our party. At the moment it all may become fatal for them, but we pass our checks and everything is fine. So let’s imagine that no one but the very special person with the very special tadpole and maybe Wyll can persuade Lae’zel and Shadowheart to put their differences aside, no one but that person and maybe Gale can deal with Astarion in a non-lethal way, and so on. As an option to make it not so harsh on players it may be just different DC for the checks. Also it may be partially based on companions classes, in such a way that we would always get a viable party.
1.3. Additionally maybe… I know it’s tons of work… but add a small level for the custom character before Avernus as a flashback of his normal day. And then a personal quest that involves someone he knew before.
The flashback level should be based on our choice of race and background. It will exist only to establish a connection between our character and a few NPC. The character has his normal no-adventures day and meets some friends/rivals/mentors. It should be short dialogs starting with the narrator characterising the NPC for us and us choosing the relationships. Like this:
N: Here comes Charles as pompous as always. Just a local thug, he never knew when to stop adoring himself. But some people think it’s just a part of his specific sense of humor, which many find amusing. You…
P: 1. …never liked this jerk. 2. … always were good friends with him, knowing that’s the humor for sure. 3. …never really paid too much attention to him. 4. …learned a lot from him.
After that the story should proceed mostly as normal. But at some point we should meet the same characters being somehow involved in the whole situation with The Absolute or any other suitable disaster.
1.4. One more big “maybe”… put some restrictions based on their temper for Origins as the player characters in single-player.
That’s the last resort and maybe probably will be hated by many players, but at the same time it’s logical. If we are taking the role of Shadowheart on ourselves, we should role-play Shadowheart with all her specifics (like hating Lae’zel and not willing to take her in the group for example). Same with each one of them.

Funny enough, I got only one type of answers, like "we don't want to be special".