The hair-covering has a profound effect on the wearer. It creates a psychological barrier, a cognitive distance between her and strangers. Her beauty becomes visible but inconspicuous; she is attractive but unavailable.
The wig achieves the desired effect exactly, because a wig allows a woman to cover all her hair, while maintaining her attractive appearance. She can be proud of the way she looks without compromising her privacy. And even if her wig looks so real as to be mistaken for natural hair, she knows that no one is looking at the real her. She has created a private space, and only she decides who to let into that space.
Perhaps in other religions modesty and beauty don't mix. This is not the Jewish view. True beauty, inner beauty, needs modesty to protect it and allow it to thrive.
The JRCC Shaitel Grant offers a grant of up to $250 towards a shaitel for a bride or a married woman who wants to cover her hair according to Jewish tradition.