Water polo in a lake? My school offered it in a heated (Olympic sized) pool, so there is a bonus for starters.
It is also a bit weird that you lump all private schools in to one group but want to highlight the good comps. Clearly Rugby is one of the A-list schools that can't be compared to somewhere with 40 in a class, just like a sink estate comp can't be compared to a top performing comp...
I spent half my education in the state system and half in the private system on a nearly-full scholarship, and I noticed really quite a lot of significant differences.
Your premise of 'all of this is available to comp students outside the school day' makes the very huge assumption that someone is there to facilitate/ferry/watch these. The private school day is 8:15 until 5pm, and is also 6 days a week. That leaves a LOT of time outside lessons for extra curricular activities without needing someone to drive you there, or juggle your preferences with those of your siblings.
How did this translate to me? My brothers and I were all able to get to county level of different sports. All bar one of those were coached as part of the usual school sports offering so didn't cost anything, and we got to compete twice a week as part of the normal school week (show me a comprehensive that would let pupils take every single wednesday afternoon off school to compete at top level…). If these coaching sessions weren't part of the usual school day, there isn't a hope that we would all have been able to be dropped off at various sports clubs in a 20 mile radius. There is no way we would have been able to buy all the kit rather than borrow it from the school equipment shed. And there is frankly no way we would have access to Olympic-standard coaches and facilities. Yes I would have learned to swim. No I wouldn't have competed at the level I did.
Class sizes were also a huge thing. GCSE classes were taught in a maximum class size of 15. My state sixth form had 20+ per class. It doesn't need spelling out what this means…
And as others have said, the private school social life prepares you for a different sort of adult life. Black tie events from an early age, visiting dignitaries where you are expected to interact appropriately, careers advice which included having alumni in to talk to you properly about their jobs (I remember Kate Aidie visiting to talk to us about the life of a foreign journalist at my private school. At sixth form college, we were given a printed leaflet…) Volunteer work was compulsory, the popular kids were the sports team captains, often on full scholarships, so status was based on achievement rather than parental wealth.
There was far more bullying of the poor kids/kids who didn't have the right clothes/kids who didn't have a car/kids who lived in the bad bit of town.
Obviously you've got such an entrenched view of how dreadful private schools are that none of this is going to resonate with you, but you surely have to admit that if there was 'no difference', a teacher at a state school wouldn't be going to a fair amount of trouble to get their high-achieving child into a top private school.