A cheap private education is a world away from the best and as was reported in the BBC article yesterday as regards Oxbridge not as good as many of the super elite comps in the SE.
Many 15k per year day schools from London and the SE consistently appear in the top 30 schools according to results, Oxbridge entrance etc. Westminster is a very top school and costs (if I recall correctly) around 20k per year for a day pupil, not 35k.
The 15k per year education my DC receive is not comparable to what they would get in the super elite comp for which we are in catchment, in terms of subject provision, class sizes, teachers experience and qualifications, facilities, careers guidance, experience with applying for top universities etc.
Of course there are some poor private schools but it would be a myth to claim that you have to pay anything near 35k to get a good one.