I admit not having read the previous 4 pages of comments, but want to contribute my thoughts as the parent of a new college grad.
I don't wish to disclose too much information about our NW, but it is north of yours and we have one child. By way of background, my spouse and I attended public high schools and graduated from college and professional schools of well-respected public universities; so we are not products of so-called "elite" educations, although we consider our educations to have provided excellent preparation for our chosen careers. It is a truism that schooling is what one makes of it.
We chose to send our child to a good, very competitive, public high school although we could have sent child to the top K-12 private school in our city. We considered that and decided against it although virtually all of our professional colleagues chose the privates school route for their own reasons. Mind you, child's HS serves a very affluent area but, because it's a public HS, there is a ethnic and socioeconomic diversity in the student population lacking in the private school we considered. Our decision was a personal one, based on our child's personality, our beliefs, and our knowledge that we would play a large role in educating and enriching our child outside of school. We had no qualms about spending large sums on travel and summer enrichment programs, which is how we chose to approach it. That's what worked for us. After HS, child attended and just graduated from a USNWR Top 10 college a week ago.
In my opinion, it is folly to "know" what is best for oneself based on what others do, even those who are similarly situated professionally, financially, or socially. There are too many personal factors outside of finances. While my child would have been perfectly fine attending the selective K-12 private school, I did not want to rub elbows with the parents there, TBH. I wanted to be an engaged parent and school volunteer, and the private school just was not my milieu. While it's likely I'd have found a kindred spirit or two there, I preferred to be among students and parents with whom I felt more comfortable and that was in the public school system, where I made many friends among the teachers and other parents during my child's schooling, particularly the early years. I know I would likely not have felt as comfortable within the environment of the private school community and that would have discouraged me from being as engaged. And that had nothing to do with high income or net worth but everything to do with personal values and choices.
Not everything was on auto-pilot for child and us during HS -- it isn't for any child or his/her parents, but our child's college prospects turned out excellent. It was likewise for many of the top graduates from child's HS. When we attended a graduation dinner for kids in child's HS class, we were amazed at the colleges some were headed to: Harvard, Princeton, MIT, Penn, Columbia, Duke, Brown and many other top schools were all represented. And that was just among the kids who attended this particular dinner.
And given that you have 3 young children, there is an additional complication of whether all 3 would be equally at home in the school system you choose. If you are comfortable with the financial commitment of K-12 private school, including fully-funding college educations for all 3, then by all means look at the private school system you find attractive. I just don't think it's strictly a matter of financial considerations.
It also occurs to me that even HNW Bogleheads would not be comfortable with the decidedly non-BH values that are more prevalent among students and parents in "elite privates," but that seems self-evident.
Good luck with your decision!