Agree with @Cantiger. Also, kids who are ultimately accepted to schools like Princeton often fill multiple buckets. Some might be a top state level athlete and national scholar in some field, others may have some outstanding skill and the solid academic record ( Mostly A’s, top courses, Top 1-2% scores, outstanding recs). There are more kids like this than you think. Then they have to spread out the kids who excel and pick the best ones. Most of these kids are coming from top schools as well. You can look up stats and see that some high schools have very few kids go to Princeton and some have a handful per year. It really depends on the school also.
Thinking that anyone is going to get in on the basis of a very good high school, moderate EC’s and test scores doesn’t help anyone. Honestly, your test scores are low and the EC’s seem to be fairly low for Princeton. I would tell my own kids the same thing if they had the same stats. I think you should reassess against your current scores ( not expected scores) and see what comes up as safety, good bets and stretches. Princeton seems like a waste.
Talk to your GC. You go to a selective private school. So the selective college admissions process is likely something very important there. If the GC will support your REA app to Princeton, go for it. If you truly want to take that shot, even if GC is reluctant, do it anyways, if you want it badly enough. My one kid shocked his independent school with an ivy acceptance. No way they pushed that app, though they did their job in submitting their part of it. So sometimes surprising things happen.
However, the reality check is the acceptance percentage of Princeton and like schools. Understand that it is a lottery ticket. It appears as though you have a well thought out list of colleges and you have seriously considered the affordability issues. Congratulations on that! You are way ahead of most kids in that respect.
Check your school’s naviance. Our private school had URM students admitted with lower stats than yours, so I think you have a decent chance. Still unlikely, but a decent chance.
Perhaps the OP thinks his URM status would help him get into Princeton. Honestly I have seen one or two URMs with similar SAT and AP scores like the OP’s got into Princeton and Yale. But that was 5 years ago, though. T10 colleges are harder to get into for everyone today.