I appreciate that the OP framed this in the context of Ivy league schools and I think that an additional year of high school has a lesser influence when it comes to those schools. Where I think it matters more is at slightly less competitive schools where top grades and advanced and high rigor classes may stand out more in their applicant pool than they would in the Ivy pool where that is the base line. Even at the best boarding schools, 2/3rds their students do not go on to attend Ivy or equivalent schools.
I think it is safe to say that the playing field isn’t level in both public and private education. To the extent that we are able, we all work within the system that is available to us. For some, that includes the option of an additional year of high school.
(I have three kids in BS and none have repeated.)