The OP (original post) describes what we
might call problems, issues, or challenges
in US formal, academic education, say, K
through 4 year college. Here is a theme
of a rough solution, necessarily "rough"
if only because US education and the
problems are not precisely stated or
understood.
(1) Learning, Knowledge, Understanding.
Yup, for doing well in life, learning, ...
is usually (being "rough" here) from
somewhat useful to crucial. Some of the
knowledge is explicitly taught in the
education, some is taught outside that
education, and still more an individual
has to pick up, ..., discover, on their
own. E.g., on my own I had to learn the
fast Fourier transform and discover office
politics.
(2) US Academics. The US K through
college emphasizes learning. In an old
joke the teacher has a full pitcher of
knowledge, pours it into the student's
empty pitcher, on tests has the student
pour back what they have learned, and
determines the grade from the fraction
that comes back.
From this "emphasis" and "process", there
can be a LOT of struggle, strain, lost
sleep, hard work, anxiety, etc. for a
student and their family.
(3) Surprise. After 4 year college, US
academics no longer much cares what the
student has learned and, instead, cares
almost only about what the student can DO,
especially in delivering new research
results. For more, do well in such
research and all the student did in
education will be forgiven, forgotten,
ignored, etc.
So, if want a degree from a famous
research university, get a graduate
degree, Master's and/or Ph.D. Generally
admission to such university's graduate
school is much easier than to their 4 year
college -- also much cheaper. For such
admission, doing well on the GRE (graduate
record exam) can help. Having done well
in an early career can help. Some
research can help a lot more, really
close the deal, giving the student
admission, no tuition, and maybe some
money for doing some ugrad teaching, work
on some prof's research project, etc.
Summary: For a degree from a famous
research university, go for a graduate
degree and there emphasize research.
Or, what US research universities care
about, more than nearly everything else,
is in just one word research.
In my case, while in grad school, I picked
a problem, for two weeks did some
research, and found a solution, clearly
publishable (later did publish the
solution). Presto, bingo, magic, I was
regarded as a star student in the
department, and everything else about my
work was irrelevant. I went ahead and did
some dissertation research, also clearly
publishable, while doing that work was
beyond any criticism, had a shiny halo,
was untouchable, got my Ph.D., and with
great joy LEFT.
The OP mentions Harvard; I got into
Princeton and Cornell but never applied to
Harvard. Sure, going to any such
university for any degree can involve more
than just "research", but might want to
critically evaluate the "more".
One more point: Research that finds
mathematical solutions, mathematizes the
subject, is relatively highly regarded.
For that approach, a big issue is learning
the math, but in grad school beware of
advanced math courses: Commonly the
teaching is poor and the hidden agenda
is to set up a competition to filter the
students. But if mostly know the math
before the course, then can be 10 yards
from the finish line in a 2 mile race, be
by a wide margin the best student in the
class, and greatly reduce the chances of
being filtered. Yes, doing so well can
cause some of the profs to resent you and
become hostile.
With math, doing the learning before the
class is relatively easy: Just get some
of the best books, usually from famous
authors, study the theorems and proofs,
work the exercises, try to find intuitive
views of the material, reasons for the
particular definitions and theorems, and
examples.
Beyond calculus, I'd suggest: Abstract
algebra, linear algebra, ordinary
differential equations, measure theory,
Fourier theory, differential forms, the
Gauss, Green, and Stokes theorems,
differential geometry, probability based
on measure theory, and statistics based on
that probability. Some good authors
include Halmos, Nering, Coddington, Rudin,
Royden, Breiman, Neveu, Cinlar. To boil
it down to just two books to study
carefully I'd recommend Halmos for linear
algebra (Finite Dimensional Vector
Spaces) and Breiman for probability.