I don't have a first hand experience but my godson goes to St Paul's Cathedral school and I got to meet some kids and parents from there.
His parents are very happy with the school. My godson started there when he was 5.5 y.o. after a year in a state primary. Whilst he was very much top of the class in his primary, it was rather difficult for him (and his parents!) in the first year at St Paul's cathedral School - IIRC it was two terms ahead of his previous primary. I remember my friend - the mother of my godson - being terribly traumatised when she saw her son's classmates' essays "What would I save in the Great Fire of London" when her son was just learning to spell simple words. The school was great in getting him up to speed though.
I remember my friend being rather concerned when the new head came on board but it seems he is very good.
Most kids go on to top secondary schools; the Head advises parents what environment would suit their offspring best. My godson who is 9 now is very bright (great in Maths and English) and the Headmaster suggested Westminster or City for him.
Don't know if I would call the parents I've met obnoxious. Most are professionals, many work in the City, some SAHM, in other families both parents work. Pushy - yes, probably, as they really value good education and want their child to succeed, so a lot of home work, tutors etc.
Don't know if this is helpful; I am sure parents whose children actually go to SPCS can provide more insightful comments.