2. I think they did the specific "not in the C of E" because it is the state church
While other churches can say "you don't believe in our doctrines, therefore we won't marry you", the C of E has to marry people who have not been married before, if one of them is resident in the parish. So I think they've put this in, even though the C of E leadership has already said "not in our churches" about this.
It would be better for everyone if it was disestablished.