It almost seems like either she doesn't really want to have a primary relationship with you, or there's something in her past that really wigs her out about meeting your family members. Perhaps she had a bad experience in a prior relationship.
I get that you're not trying to cage her--she's special to you, and you want to share that with your family, and share your family with her, because they are also special to you.
It's worth talking about, perhaps with the aid of a counselor. If you're not going to take the counselor route (perhaps owing to cost), at least read the NVC book together and practice those techniques. Emphasize that you're seeking to understand each other; you're not seeking to "win" in an argument.