Even some private gifted schools cannot accommodate PG kids. That's the ugly reality, or has been the case with DS7.
DS was in a traditional private gifted school for pre-K at 5; it was structured and classes were grouped by a couple of ages. DS accelerated through the pre-k/k/1st grade curriculum within 2 1/2 months. Headmaster refused to grade skip him to 2/3rd grade class or even just for math.
I then put DS in a Montessori-type private gifted school for the remainder of pre-k and k. This was fairly unstructured with kids from pre-k to 8th grade. DS had the 'celebrity' thing there too. Teachers said that he might not be there long.
Well, this year I'm homeschooling. I've been using a general curriculum guide series for some structure and to make sure the 'basics' are covered. I started at grade 1 and DS is now in Grade 5 and likely to finish the series at Grade 6 by the end of the year. He's also reading adult level books on the Mayans, Incas, Ancient Egypt, archaeology, and English history.
What I notice is that the PGness might not be so glaring in the company of others, depending on the child. My DS is an introvert too so he doesn't always show his true colors in public spaces or certain settings. Last year, he took the WJ-III and did not receive a qualifying score for DYS either.
Frankly imo, I don't see how a private or public school can really accommodate some PG kids. For one, I don't think the schools physically have the books or materials that some of these kids seem to vacuum. I just don't see many elementary schools having the expertise or resources on particular subjects and topics. The vast majority of kids have to be coaxed to read a book on the Mayans or Incas; instead of having a child who has gone through the public library's children's and adult collections of it

. And that's just an example of one subject and one rapid acceleration.