I know Princeton pretty well. If my children wanted to do pre-med I would strongly discourage them from going there. The grade deflation is waning but you still will be competing for grades against future top-30 professors and the like. It’s very tough.
At almost all schools including competitive privates need to be at the very top of your graduating class (top 1-3%) to have a chance. Private school scattergrams tend to be skewed by prior applicants with hooks, and counselors aren’t always willing to be straight about that.