I dunno, your thinking is a bit skewed here.
Clearly you have made the assumption that all private schools are academically competitive (they're not) and that all parents choose private for grades (they don't).
You also seem to think that all high-achieving kids from comps must feel like they've 'won the lottery', but why? They are likely top set kids who have worked hard and knew from their predicted grades what was possible, plus knew what they had to work towards in order to meet entry requirements for their first choice course/uni.
Unis have been increasingly pro standard secondary education in recent years. Many kids now leave fee paying schools for A-levels to increase their chances of getting into top unis. Oxbridge particularly. We no longer live in a World where private kids snag the best uni places.
At the end of the day, grade boundaries are the same for all and kids are only capable of what they are capable of. Many of the perks seen in private are experienced by top set kids in comps - smaller class sizes, best teachers in exam years etc.
Many privately schooled kids are self-driven, self-motivated and resourceful too. Our local private is a boarder and whilst those kids are getting a great education they are not experiencing the privilege of an invested, loving family supporting them every day.
You just seem a bit, well, salty? Sneery?
I'm really not sure it's necessary but let me tell you, as a parent of a child who did well on the 11+ and got a place at grammar, I have experience of your attitude.
I know for a fact that when GCSE/A level results come out the parents of his primary cohort will be VERY invested in his grades and GLEEFUL should he be seen to have failed. Why? because their kids didn't get in and they are perpetually pissed about it.
Fair? No. Why? Because kids are kids. Private kids may be getting an arguably better education but they still experience life - illness, death, mental health struggles, abuse.
There's really no need for the boasting, IMO.