I can only speak to the East Coast schools. Every single school you are looking at and that everyone else has mentioned would meet your requirements. One of the hardest things about this process is there are so many options to look at.
I first try to narrow it down a bit in terms of type of school: coed or single gender? religious affiliation (and, related, mandatory chapel)? Saturday classes? dress code? how strict? 100 % boarding or mixed? Rural, suburban, or urban? Within that find options that competitive and maybe less competitive.
Then look a bit at culture. Some kids love and thrive the “elite prep school vibe” and there is nothing wrong with that. Some are move looking at a place to explore and be creative about who they are.
If the top schools in the Northeast we found Hotchkiss and St. Pauls to be less on the nurturing side than some others although they are both amazing schools. We ended up with Loomis because it had more of a “family” feel and the kids don’t seem as competitive with each other. My son loves being able to walk into a town and get barbecue or breakfast with other students, but some don’t love the high number of day students.
Our friends are LOVING Milton, which is in a Boston suburb and which they find very supportive. NMH is great for a more groovy vibe and no dress code but gets a bit less respect within the prep school world. Berkshire is a wonderful hidden gem but, like NMH, is very rural and isolated.
Good luck and have fun.