I think a lot of this is because sport is valued by the private sector in a way it just isn't in the state sector - and actually all 'public success by old boys/girls' matters more to the private sector - it's part of the marketing.
But also it's a culture of the public schools that's just not there in state schools - state schools are judged solely on their ability to get DCs GCSEs and A levels - their ability to teach the subjects they have to teach well. Success in sport, drama, music, are 'nice to have' not 'vital to our school success'.
Private schools do have more emphasis on the "whole child" education- you find that with the debating clubs as well as the sports - you will find over and over again comments about the confidence of public school boys/girls, this isn't something they just magically got, the schools put effort into this even though it's not going to get you a qualification.
Private schools are judged by parents, not just by league tables (although those are also important) and most parents aren't just paying for good results, they are paying for all round education.
State schools are measured and controlled so much more on things that can be measured on that year group - quite frankly, 'wasting' important learning time and limited school resources on minority sports on the off chance that in a decade time, long after leaving the school some of the DCs will get olympic medals is a huge waste of time for a state school. (Unless you find a teacher with a passion for a particular sport).