Sometimes the schools care.
Over a decade ago, one of ours was interviewed at a prestigious private day school by a prized science teacher. When the school asked why ours elected to go elsewhere,we shared one remark that the teacher had made during the interview. Nothing shocking, but the school let the prized science teacher go. (Not sure whether he was fired or non-renewed contract.)
The school noted by a couple of posters above in this thread–not Hotchkiss–is often impressed by itself & could use some critical feedback.