The financial aid I'm talking about is offered by some Ivies and other extremely selective American universities and liberal arts colleges to international students demonstrating exceptional academic potential.
Most students at the Ivies, etc., do not demonstrate exceptional academic potential, unless you consider top decile, or so, to be exceptional. At these schools, there is no expectation or intent that all students should become academics. Your chances of gaining admission go up with academic achievement up until a certain point, which reflects the level that the university expects to teach its classes at, but then plateaus until a child really does get exceptional. A few students will be admitted because of truly exceptional academic potential. Most students will be admitted because of other demonstrations of potential.
The financial aid offered by these academically elite institutions is based on the demonstrated financial need of the admitted applicants.
That is true. They have almost all shifted from merit-based scholarships to need-based aid, on the theory that merit-based scholarships tend to reward students who were already advantaged by having a lot of resources to develop their merit.
Your rowing prowess (or slight stature, ability to swim, and willingness to learn to cox) might get you onto a crew once you arrive in Harvard or Princeton or wherever, but it won't get you in in the first place,
That is not true. Once a certain minimum level of academic achievement is demonstrated, Harvard and Princeton look for demonstrations excellence in something, and that something is usually not academic. The idea is that someone who has the discipline and hard work to compete at a national level in, say swimming, and still maintain acceptable academics even after 20 or 30 hours per week of practice and competitions, will be able to keep up at the university and will eventually find some other area in which to apply their discipline and hard work and, hopefully, excel.
and it won't pay for your four years there.
That is true. You are not obliged to continue your sport or music or any other activity to maintain your need-based aid. You are, of course, required to achieve sufficient academic results to remain at the university, but these requirements are nowhere near 'exceptional'.