I think it is very rare to find a school, private or public, that will challenge a gifted kid.
I’d go so far as to say that you might have to go with “least visibly damage”.
In my experience (I am not in the US, so take it for what it’s worth), it’s not so much about public or private but about whether the administration as well as the actual classroom teachers are flexible and accepting of the concept of giftedness both in theory (as in, yes gifted kids exist, and yes they have different needs as opposed to “all kids are gifted” or “we don’t believe in giftedness”) and in its practical manifestation as in, they accept yours as opposed to “she’s not as smart as you think” or “some kids may be gifted but I’ve never had one in MY classroom” or “all the children in our school are so advanced it won’t make a difference”.
Private schools we visited (and yes, that included a Montessori school) tended to actually be less flexible than public schools in outlook (“this is what we do and how we do it and we are great at it and will work for every gifted kid, too”) even though they had better resources and fewer children in the classroom whereas public schools were more like “we’d love to meet these kids needs, but we just can’t.”
We went for the public option for preschool, a parochial option for elementary and a public option for middle and high school again. Each time, we liked the way teachers and administrators were openly accepting of both the idea of giftedness and the idea that our kid might be, and honest about their limitations.
You may not be able to make academic growth a priority, but I’d say it isn’t strictly necessary. One poster once put it “if he’s happy, engaged and learning, it might be enough.” As long as he’s not stunted, belittled, bullied. An emphasis on things such as art arts and athletics might actually be fine because it does not force kids into academic lockstep or isolation. You can always introduce academic challenge at home and bide your time until more challenging school options open up - the important thing is to avoid damage.
