You miss out on a personal storyline centered around your character. That's some pretty substantial content, I think. Sure, you can see Shadowheart's etc arc from the outside and influence how things turn out, but that's just the same as every other character who isn't Shadowheart. You'll be playing through the game only getting the 'default' story beats That other characters get when they don't have special content in a given scene/dialogue etc. But for *every* interaction. It's not as if you can even say 'yeah, but I'll roll a dwarven bard, we don't have any bard or dwarf companions, I'll have a unique time there!' Because Dark Urge is also an option. And Dark Urge is fully capable of having any class/race combination, while also having their own storyline.
You'll be missing out on a ton of content. Hours even, I imagine. Your Cleric of shar Tav will never immerse you in the story as much as playing Shadowheart. Your Githyanki fighter will always feel like a pale imitation of Lae'zel etc.
God I hope Larian proves me wrong, because I really don't want it to be that way.