Sorry, OP, but your post is a good argument for why distance is a "fair" means of selection.
Why on earth should people be able to utilise the various accoutrements of privilege to wangle their child's way into a school? (joining the PTA, helping out on school trips, etc,) Why shouldn't girls (and I notice you don't think the single-sex criteria is unfair,) in a "less-desirable" area have first dibs on a "good" school?
There are arguments against distance/catchment selection, but, in this instance? Sorry, I'm all out of sympathy.
Seriously, the only thing to say is that the unfairness resides in the disparity between the schools on offer, which is what makes the mode of selection, amongst all the parents who have recognised this disparity, and understandably, would rather their girls went to this school, an issue.
As everyone has said, you now have the option of accepting the place you dd qualifies for, and if necessary supporting and supplementing that (not fair, agreed, there will be lots of parents for whom those last two are not an option) or moving (again, not fair; lots of parents can't do that).