Bro, muslim girls have not come down to Earth from a distant planet.
If the rest of female-dom on this planet does not have this overpowering desire to be a recluse for life by hiding ones perfectly unblemished face from the world, then it stands to reason that the only reason muslim girls are doing it is because of something that is isolated to their religion. Because it crosses cultural, ethnic, national, and continental barriers.
So that answers your question about them really wanting to do it. Versus having to do it.
You think a young girl NEVER asks herself even once the first time she is made to wear the veil, why her brother does not have to wear one? Ask yourself that.
You think girls LIKE being covered head to toe in black cloth walking in the afternoon sun? Ask yourself that.
You think girls LIKE seing their childhood friends moving on with their lives, without a veil, getting careers, having boyfriends, having a life - while they move around in those voluminous gowns waiting to show their faces to one man for the rest of their lives? Ask yourself that.
Now ask yourself WHO the veil really helps.
We are not even getting into the security angle here.