First, I don't have any kids who went to KOR. I did have kids at St. Thomas Early Learning Center years ago and it is a group of those parents who started KOR school instead of sending their kids on the public school once they finished the STELC after first grade. Eventually, KOR added kinder and 1st even though those grades were covered at STELC.
The KOR school has very involved parents. Christian values. It is a four day a week M-Th, half day program with some extracurriculars on Fridays if you wish. I would not say it is like any of the other private schools in town---more like an extended home school program. Small class size. Interested, involved parents. Challenging curriculum.
It does not surprise me to hear that KOR kids would do well in contests and such but I'd think those kids would do well in almost any environment.
Finding the right kind of school for your kids is important so good for you for asking questions. Good luck in your search.