If you are a Robertson Scholar, I am assuming that you are a driven high achiever. Assuming that drive and determination follows in your college career path then all the schools you listed including UNC will give you a world class education with the opportunities you seek.
Will an IVY school open doors a UNC degree can’t? I honestly can’t say. Some say it can for investment banking and some select other Wall street type jobs. I can say that at the end of the day, most doors are opened through your hard work and accomplishments. The name on the paper only goes so far and generally only for the first job. After that, it is all about what your job experience is. I believe that the Robertson Scholar program can provide the opportunities you need to reach your goals if you seek them out.
A few random thoughts:
I strongly recommend you focus the quality and reputation of the program. A school with great prestige is worthless if the quality of the program is lacking. A MIT degree is impressive as **** but is it worth it if you want to study history (I have no idea. I am guessing history is not one of it’s better known programs). The schools you mentioned will have strengths and weaknesses and different personalities. Some are more cooperative and some are more cut throat. Only you can decide what works for you. Some thrive in a cut throat competitive school. Other don’t.
Do not kid yourself that UNC will be a walk in the park. Your fellow Robertson Scholars as well as many other students will be as impressive as you are. You will be challenged and maybe even humbled. My daughter has meet many in this program and felt that they were a sharp group of good kids.
Don’t be too concern too much about class size. Yes some are large, particularly the general education classes. I assume that you have a boatload of APs, so you will likely be able to skip most of you general education requirements with those. Because of this you will be able to take more courses in your field or devote more time to research. This is often not the case at many private schools. Also by picking honor’s courses you can often get smaller classes.
My daughter had no problem getting to know her professors. Many knew her well enough to give strong letters of recommendations. During her four years, all of her professors (with maybe 1-2 exceptions) were full time PHD professors. She was not a CS major so this might not hold for you. This is not necessary true for other schools. I know one IVY has a lot of graduate student teaching undergraduate courses. Not always a bad thing. Carolina is also still known as a school that supports the tenure track with less reliance on adjuncts. They will also hire professors that only have teaching responsibilities.
While my daughter did find the research opportunities were limited in her field, she was very successful in finding related research opportunities outside her major. This led to a great job opportunity in the field she wanted.
While the IVY schools can have strong networks. You will find Carolina graduates everywhere.
One Robertson Scholar I met made a interesting comment. While UNC is a large school, you can make it as small or as big as you want. He had a very specialize field and end up with a tight group of student with common interests. He was a Physic major who was accepted to MIT but choose UNC. I got impression that it was a money issue but he definitely made it clear that he had excellent opportunities at Carolina.
I suggest that contact the people running the Robertson program and see if you can talk to some students in the program.
If you are really concern about prestige then Carolina might not be the school for you. Not because it doesn’t have prestige of it own but going to a school that you feel doesn’t meet your definition of prestige can color your college experience. Some can move on but other just can’t.
For most jobs, the name on the degree does not translate into a higher starter salary or the better assignments. You will be doing the at same work at the same salary as the graduates from the non-IVY assuming you have the same skill set and experience. After that is all about performance.