I disagree with private schooling politically, so we choose not to privately educate ours, but it's a truth universally acknowledged that parents and their overall involvement have a bigger influence on educational achievement than type of schooling. If you are there for the before and after school stuff, you can add far more value to them educationally, by helping them with homework, taking them to extra curricular (non educational but still really good for development!) stuff, and providing a stimulating, interesting and interested environment instead of them being in (a no doubt very good) after school club at their private school. After school clubs are great, and I use them myself, but no one can deny that a stay at home parent is the best possible care if there is a choice. Mummy is better than granny is better than nanny is better than childcare, but for the older child. Not everyone can do it, of course, but its value should not be underestimated.
My non privately educated sister and I got straight As. We had very little money and did no additional after school activities. We were capable and had interested but not over involved parents, who left school work, homework and exams entirely to us. However, they provided a rich, discursive environment for discussion, challenge and thought, in every day life, for the first 18 years of life, which is far more important for overall achievement than how much your parents earn (and therefore whether they can afford to throw some money at it).
If you have outstanding schools near you you are very lucky and should grab the opportunity. Stand your ground. I like someone's ' we don't need to pay for schooling because the schools here are so good!'. That should shut them up.