My thought is that knee-jerk responses to just about anything are usually short-sighted. The right choice varies by kid, by family, by location, and even over time.
I was a public school kid and am very much in favor of the socio-economic diversity. I wanted a gorgeous house in the city, but it would have meant 2x the property taxes + $25K+ per year for private school, so the math very clearly supported the suburbs. Found one with good public schools, so we were set.
Then I had a highly ADHD kid who thrived in a Montessori preschool. So we kept her there; she's one of those wicked-smart-and-wicked-immature kids, so she was very much a handful. The independence and ability to move around and self-direct was such a plus, I couldn't see changing schools when she hit K.
Then the Montessori school switched up to more like Asian Prep Academy -- very rigid, lots of worksheets, and I said, well, if I want a once-size-fits-all approach, I can get that for free at the local public school. So we switched her over, and lo and behold, they were better able to handle her behavior issues -- they put her in the class with the teacher with two ADHD sons, and poof! she was suddenly completely manageable. She has had a great, challenging education there (e.g., took Calculus AB/CD as a junior and is now in Multivariable Calc as a senior), and I have no regrets.
But now that we are looking at colleges, we are looking both public and private, and she will go to the best fit that she can get into. The first factor is that our local publics are not great -- they don't do particularly well in her area of interest, the campus sucks and is known as a party school, and they have developed a strong reputation for forcing kids to stay a 5th year because they can't get the classes they need to graduate. Not to mention ADHD kid + overwhelming party school = recipe for disaster. So once you consider 5 years or more to graduate, and the lack of support/likelihood of a complete crash-and-burn, public doesn't look so hot.
And then once you start looking out of state, surprise, the tuition difference between OOS public and private is nonexistent (U Mich. is something like $60K+!). In fact, one of her favorite private options is significantly cheaper than many of the publics, because they have a good endowment and give half-tuition scholarships to everyone. So we are looking at all of the options.
But the real key is that we are also looking at schools at all different levels -- reach, match, schools where she'd be at the top of the class, etc. From seeing friends and relatives go through this, it is clear that many private schools will offer merit-based aid to kids they want to attract -- my niece had a number of offers from both publics and privates that all brought the cost of attendance down to the cost of an in-state public (gee, what a coincidence that they all came up with the same basic number, right? Almost like they've figured out how to attract kids from out-of-state or something). So it seems very likely that some of the lower-tier schools will come up with significant scholarship offers -- and some of those schools also happen to be very good in the area she is interested in, so win-win.
She may or may not choose the school with the best financial package; honestly, we have known that she is, umm, "unique" since she was about 30 minutes old, so we have been saving to afford the college that was the best fit for her since, oh, about the day after that. But the larger point is that if you have a smart kid, it is flat-out stupid and short-sighted to just ignore private schools or out-of-state publics, because the ones that want your kid will come up with the cash to bring your costs down to the in-state level (or better).