First of all, yes it is possible for you to get accepted to a different highly ranked university. In fact, many years ago exactly that happened to me when applying to graduate schools. I was rejected by Cornell, and a couple of weeks later got accepted to Stanford. Years later I finally realized that Stanford had always been a better fit for me.
Top schools may differ in terms of how much they care about your freshman year of high school. They also are each looking for students who will be a good fit for them. You might have the same result that I had, where a different highly ranked university might feel that you are a good fit for them.
However, if you are intending to major in almost anything else other than investment banking, prestige will not matter much and you can do well at a very wide range of universities. If you were to graduate from a university on the Yale, Stanford, … level, then after graduation you will find yourself working alongside graduates from a very wide range of universities and no one will care where anyone got their degrees. At least to me, anthropology and environmental studies are areas where some graduate school is possible. Again if you look at students at the top graduate schools, they will have come from a very wide range of undergraduate universities.
And you have some very good in-state public universities in New York.
I also wondered about this.