Following this post!
I could have written this myself. Last year, my ds was in 1st grade as a 6 year old. He had one behavior issue after another. Finally I decided to get him tested: IQ, ability, inattentive, & ADHD. His test results showed that he was very bright - gifted. My son was at a college prep, Independent school, so I thought he was being sufficiently challenged, especially being the youngest in the class. Turns out there is a very high percentage of students at this particular school that need special help because they have diagnosed learning challenges and disabilities. So the teachers are spending much more time and resources on those children to help them get on-level. My child got extra enrichment packets after his regular work. He saw it as a punishment and it wasn't challenging for him, just more than what his friends had to do.
My advice for you is to ask the private school that you are thinking of paying $25-30K a year specifics about gifted curriculum and the percentage of "gifted" children in your son's grade level. Many high-ability and gifted children can be ignored because they are performing as they should. When they feel ignored, bored, or injustice, they tend to misbehave or do exhibit behaviors that can be disruptive.
Best of luck!