We sent our S to (supposedly strict) private school for K to learn some structure. Big mistake - turned out to be too much structure, too little flexibility (even bathroom trips were scheduled on the fixed times - for 5-year olds!). Public school for his 1st grade: wonderful teacher, relaxed yet structured atmosphere, learning 2-yrs worth in a year. After that we didn’t experiment with private schools any more for both S and D.
However, the rest of the elementary and middle school for both S and D was pretty bleak. Sure, there were some good teachers but they couldn’t do much because they were required to stick to the district-approved curriculum (and no grade jumping!) and my kids (along with a good half of their class) were well beyond that level. It did get much better in (public) HS.
Another anecdote from a relative with a kid in public K: a boy is extremely talented. They were doing some very simple additions at school. After doing some thinking at home, the boy “invented” multiplication! He was drawing some other symbol for a sign, though. The teacher at school not only completely failed to see that the boy is bored to death (adding 1s and 2s for the whole year), she referred them for psychologist to see if the kid is ready (!) for 1st grade because he is not paying attention in class - like there was something worth paying attention to.
My conclusion for our district (a very good district feeding into HS which is top-of-the-line) is if you have bright kids (I don’t even mean geniuses - just bright) and IF you can afford it, consider either private or homeschooling for elementary+middle. Choose the private school carefully. Regular classes in public schools go so slow that there is barely any learning going on except for how to deal with boredom. Especially beware the “common core” curriculums in public school districts - they are even more watered down to the level no sane person can imagine.