_Morpheus, as a teacher in the United States for many years now... you are the one that is incorrect. Students apply for and receive scholarships very early. Not all scholarships come from universities either, but even universities start searching for talent early.
We have had high school students as young as 9th grade do so well at the science fair that they received a scholarship. Universities and business corporations often sponsor these events to search out talent they can woo to their university or company. Also, not all scholarships are large amounts or full ride scholarships.
There is also the Duke Talent Search that starts as early as 4th grade, but the famous "test" is given in 7th. I don't believe that Duke University grants scholarships through this program, but it definitely puts kids on the right track to possibly receive scholarships early in their educational career.
If anyone has kids, I highly recommend starting the scholarship search early. It is worth the effort of filling out a gazillion applications. You don't even have to be that academically gifted. Even average kids can garner quite a few small scholarships sometimes.