Hate to say it but I would be telling that dentist to go to hell. I am in the room for everything for my boys except for the boy part stuff and then I make my dad go with them. But either way they are never without a parent or grandparent. Your little girl is only 4. If she doesn't have mommy or daddy in there she is going to be scared. And how do you know that they aren't going to threaten her or something if she gets out of line??
Im sorry, I am overprotective too and I trust NO ONE with my kids.
Stick to your guns.