Your Summary is Spot On

Most private schools are looking for kids with good GPA + great ECs. You said that your child did not have leadership and great recommendations. My guess is that the teachers are very polite by telling you that your child is shy but smart, which basically means the rec letter was just OK, not exceptional.

I personally don't think going to ivy school is the ultimate target. However I do think that it is helpful to teach our kids on how to interact with teachers and leave a good impression. Kids who mature early always have some advantage in this aspect.

My own child is attending Penn. From what I learnt, academically there is not much difference between ivy schools and good public schools. Eventually it comes down to how your child can utilize the opportunies each school provides. Penn is hyper-competitive in my opinion and very stressful. We are middle class, so sending a child to private school means a lot of financial burden. My kid knows this and I think this put stress on her, because in private schools you meet kids from super rich families. In this aspect, I do see the benefit of going to public schools.