MN164, I think you are being a bit unfair. The gown and the nomenclature come with the job, and not obviously the aspects Westminster's current Head prioritises. But I agree with the Hogwarts analogy. Boys often seem to know which of the two different environments they prefer.
That said you are right. Any choice now, essentially includes the choice of 13+ school. The occasional WUS pupil might go on to board, but very few will switch from the Westminster system to St Pauls or vv.
I also disagree with Enormous Tiger. Lots of boys at Westminster have SPGS siblings, especially younger siblings. A surprising number of Westminster sixth form girls are siblings, so presumably the thinking is that by selecting Westminster over St Paul's for your son, your daughter has a better chance of accessing a top notch co-ed sixth form experience.