There are private schools, and then there are great private schools. Our daughter went to a private school in Georgia, made in the mid-750's on all of her SATs and SAT IIs, and went to MIT. We thought she was golden. Unfortunately, what we now understand that we didn't understand before is that for bright child to be happy and unstressed at a top 5 university (a) it takes more than being very intelligent and having high SAT scores, and (b) not just any private school will do. The child must be *educated", not just smart. The kids at MIT who were unstressed were the ones whose schools prepared them explicitly for MIT and the Ivys. I didn't see any McCallie kids listed in the MIT facebook when my daughter was there. From what I see, they mostly end up going to schools like Emory, Vanderbilt, or Sewanee, great schools, but not MIT. Many of the kids at MIT, for example, went to schools that use the exact books that MIT uses for its freshman. So when these kids get to MIT, they've already worked all of the p-sets (problem sets), and they breeze through with little stress.
BTW I totally underestimated the power of being a legacy. Legacies don't get any preference at MIT, but they do at the Ivys.
If I had to do it over again, I would have sent our daughter to a private school of that calibre. Oh, well, too late now. For an average student or below, I'm not sure that it matters - might as well save your money and put him/her in public schools. For an above average student, I think you'll get your money's worth at McCallie or Greenhill or private schools of that calibre. But then I've never crossed those bridges, so this is just speculation on my part.