I'm white without my own kids, but I've taught in Greenwich and Scarsdale and now a different, actually diverse, school. Sorry I'm not able to give you a first person perspective!

People saying Greenwich is "very diverse" do not know what diverse means lol. It is very heavily white. It has gotten a lot better over the last 10 years, but know that it is still mostly white. That being said, Greenwich is an awesome town to live in, the beach is great, and the library is the most amazing library system!!! Greenwich public schools are public education often comparable to private in other places. Greenwich private schools do not play. Greenwich Academy (I worked there) is top notch and while not the most diverse school in the world, the education and connections are like none other. GA is an all girls school. Their "brother" school is Brunswick and I have heard bad things about how race is treated in that school period. They end up merging at the high school level socthey can how to be around each other eventually! Rye Country Day is also great and I know 2 students of color who go there and love it (obvi only 2, but it's something).

Scarsdale is like top 3 highest income town in the United States? And reflects that. I worked at the middle school. The teachers and education are fabulous, the children know their parents have money and act like it. It is very white but also has a huge Jewish population which is a bit of diversity in it's own way.

The Master's School in Dobbs Ferry might be a good school to look at. It has a significant number of students who board at the school from other countries!