Will going to private school effect my chances of attending an Ivy League or other elite University?

Correlation may not = causation but there there is too much here to dismiss. You can’t prove smoking causes cancer either with stats, but there is a pretty strong case if I recall my epidemiology.

Many long time posters seem to put Ivy admissions deans on some kind of “all knowing” holistic pedestal. I know there are quite a few hacks all over higher ed. Not an Ivy but check out this clown:
https://www.timesunion.com/news/article/SUNY-official-placed-on-leave-after-article-12930446.php

The beauty of being a hack in higher ed , who would ever know until you really screw up.

Last year when we had the opportunity to talk to a Harvard admission rep and my kid asked about Math 55 trying to engage her in a discussion. She had no idea what we were talking about. How is that even possible? She was a H alumna.