I don't thing it is impossible. I'll expand. For what I've seen in DOS and BG3, the characters created by Larian like Astarion have labels to identify them. Those labels are linked to their quest, stories and some dialogs.

So you can basically write some stories asocated to quest and a few dialogs and give them labels. Then you let the player creating a custom character choose one of those stories+quest. If you want that the stories make a little more sense with the custom character and felt a little more coherent, you can restrict the selection to choose from depending on race, class and background.

For example: you might want that the story "dwarf heir" only show to the people choosing a dwarf.

So there is a potential technical solution and I would like to see something like that. If that doesn't happen well, it won't be the end of the world. But I would really like to see that custom characters are as rounded as the ones they created. Just as they promised. Hell, they might be already discusing better solutions than the one I said. After all, they are professionals.