One of my clients, a UN official, was kind enough to email me this today:

I totally agree with you, as you said “In my experience, international families go public if their children are elementary-school aged, they will only be here for a few years, and they don’t want to deal with private school admissions requirements or fees.   However, on the flip side, affluent international families who want to stay in Manhattan often seek the most elite private schools they can get.” 

I should say that, it is the general feeling among my friends, including many UN colleagues, to consider sending their kids to the private school starting from the age of Middle School if they decide that their kids will go to US colleges. The primary consideration is the quality of education and the effective college admission assistance.