Here’s the thread that I wrote back in 2011 after we had gone through process with older daughter.
I do think St. Andrew’s should be on your list…arguably above a few of the East Coast schools you list above. For starters, like SPS, it’s 100% boarding, so the “immersive” thing is there. I also think the ethos you sound like you are looking for is very much a St. Andrew’s thing. Another school to consider is The George School in Newtown, PA.