All of the schools on your list are peers and each will provide everything you say you are looking for. If your safety concern about SPS is referring to the Owen LaBrie/Chessy Prout case, Owen is no longer there, and each of these schools (and all the rest, public and private) have had their scandals. None of the boarding schools condone or tolerate this behavior. You are no more or less safe at any of them in that regard (you will be safe), so I think you can set that worry aside. You and your parents can read some of the discussions from the archives on this topic:
Sex Scandal and Boarding School
Senior Salute and Other Boarding School Traditions
I’m Beginning to Feel Scared to Leave for School
When the Sheep Dogs are Preying on the Sheep
But don’t forget to look up the scandals at other schools, too:
I’m Trying to Find a Good Boarding School but All of Them Have Scandals
Alcohol and Drugs at Boarding School
…and the list goes on. Most of these incidences are pretty old, but no school has an unblemished slate just, perhaps, an unpublished one.
“Large” is relative. Exeter and Andover are the largest boarding schools, but each grade is only about 300 and classes are very small. That seems very intimate to me. There aren’t many stories here of students falling through the cracks or not being able to find their herd. Choate is also one of the larger schools but everyone knew almost everyone at least by sight if not by name and, within grades, the communities were known and tight. There is no need to know everyone at your school. No matter the size, your personal friend group will only be a handful. You will not get lost at any of your choices. However, if you prefer a smaller community/campus, that may help you narrow your choice.