There are a few ways an individual can obtain an "automatic" appointment - being the child of a CMH winner is one, and I cannot remember any others - but for the most part appointments are made only after one has been nominated, and then had one's nominations carefully and thoroughly vetted by the Academy in question.
Is that correct?
Yes you are correct. Check the attached links for both USMA and USAFA processes. There is a category for ROTC honor students, which is probably what Carson would have been competitive for, but it is patently obvious to me that Carson has been pushing a fabricated tale for years. The Carson supporters are in overdrive trying to spin this as a positive, but Carson’s words are there for all to see.
http://www.academyadmissions.com/admissions/the-application-process/nominations/
http://www.usma.edu/admissions/sitepages/apply_nominations.aspx