or wanting to send their children to a faith school but complaining about the faith content of the faith school. But there are grey areas as with everything. Is that christian privilege?
Try actively not wanting to send your child to a faith school but having no other option for a state education because faith admissions have so skewed local catchments. 
The only reason we got into our local one rather than being allocated to the failing (CofE) school on the opposite side of the borough was a less-than glowing Ofsted that suddenly saw the parents who felt they must have a faith education even though they live some distance away suddenly, er, re-evaluate their priorities....
I happily acknowledge that I am aggrieved about this.
However back to your question - I tend to count as Christian privilege that bestowed by the state that is mostly or totally denied others/none.
The weddings/christenings one is a hangover of this - the weddings, anyway, because of the special administrative status the state confers upon the CofE to perform marriages.