A child can live next door to a school and not get a place because a child from a family of faith living 5 miles away has taken it. Is that fair?
You could put in that sentence any admission criteria. The child living next door to the school can lose out to LAC, siblings, etc. I would say it is no more unfair than that. You could use this argument for any admission criteria of which you disapprove.
As I said above my local catholic school is about 50% faith admissions
As t4nut indicates, faith schools vary. Some, usually in areas where there is a shortage of places at good schools, tend to fill up with faith applicants unless they reserve some places for non-faith applicants (which is unheard of for Catholic schools but happens at quite a lot of CofE schools). Others regularly admit quite a high proportion of non-faith applicants.
calling someone a liar with no evidence is pretty low
I would not call her a liar but she is at the very least mistaken. In the normal admissions round you apply to faith schools through the LA in the same way you apply to any other school. She could apply to her local faith school and it would have been considered. That doesn't mean she would have got a place, of course.