Or let's put it another way.
You have resources : time, energy, commitment to your family, being capable and emotionally strong.
You use the resources you have (which not everyone has) in many different ways. Some of those ways will benefit your community (like having a sports club they can make use of). But naturally most of the benefit of your personal resources accrue to your own family. You made the club because it's your DC's personal interest. Your DC benefit from your abities in many ways to have a wonderful childhood. This makes them incredibly privileged and much more likely to have a good life than a child whose parents don't have equivalent resources. If they also went to PS, that would benefit them - but they have enough other privileges that it doesn't matter.
A private school parent (me) also has resources, including a high paying job. There are other resources I don't have, notably time and energy.
I use the resources I have in many different ways. Some of those ways will benefit my community (like the large amount of tax I contribute and the coaching I do). But naturally most of the benefit of my personal resources accrue to my own family - just as yours do. I use my earnings to give my DC the best education I can and my DC benefits from my personal knowledge, commitment and access to activities I provide to have a wonderful childhood. This makes her incredibly privileged - but not any more so than your own DC.
There's really no difference between the privilege I give my DC with my resources and the privileges you give your DC with your resources.
Neither are bad - we're meant to try to help our children become good adults. It's our responsibility as parents. We have different resources, so we do it in different ways.
The privileges you give your DC - a secure home, supportive parents, access to activities which help them grow socially and developmentally - are much, much more significant than which school they go to. All the studies of childhood outcomes show it.
Stop being angry about what other people have that you don't, and appreciate what you do.