I think the reliance on letters of recommendation for admission to graduate programs in the U.S. educational system (which is not at all homogeneous, and is not really organized in any way), is substantially due to the general breadth-but-shallowness of all (to my knowledge) undergrad degree program requirements, plus the wildly varying resources of all the various colleges and universities, plus the wildly varying opportunities (due to both the college/university and students' socio-economic situations).
The typical (maybe not most-elite) U.S. grad programs in math allow people 5 or 6 years, explicitly acknowledging that many students will need to learn a good bit more... since in most cases they simply did not have time/opportunity to do more math coursework as undergrads.
Yes, often, getting an M.S. in math in the U.S. compensates for this, if one can do the M.S. at a reasonable place. Oop, but most M.S. admissions (currently, in the U.S.) do not include funding. So, if money matters to a person (and, to some, it simply doesn't! a colleague of mine chronically expressed bafflement at faculty concerns about salary and benefit... which was eventually explained by discovery that he was an heir to a billionaire), a paid-for M.S. is probably infeasible.
So, one would/should want to be admitted from B.S. to PhD (and maybe, in effect, get an M.S. along the way), because most admissions to PhD programs pay tuition and a nearly-livable stipend.
Simultaneously, applicants from nearly every other country in the world have been operating under a different system, so that their B.S.'s are U.S. Masters', and their Masters' are a fraction of a PhD.
So, perhaps, by some formal rationale, essentially every U.S. student fails in comparison to non-U.S. students.
Perhaps so, in_the_short_term. But the short term is not the criterion of most interest.
Yes, it would be reasonable to speculate that a head start gives a permanent advantage. But, very-interestingly, this seems not to be the case (in my observation). Sure, in some cases, but, by far, not reliably so.
So, how do we compare apples and oranges? I think that rhetorical question is a correct explanation/analogue of the issue of comparing students from different educational systems. (Which is a legitimate version of the original question. If, as in some comments, it's about "asian-americans" versus "anglo-americans", then I have nothing to say!?!)
So, how to gauge the probable successes of people with different starting points, and with most U.S. candidates having so little tangible experience that it is very difficult to assess their talent for mathematics based on coursework grades and GRE. Not to mention my "very-mixed" feelings about typical undergrad U.S. math curricula actually giving an idea what live math is about.
(I have to add that I'm glad I was able to "test out" of essentially all undergrad math... partly, indeed, by being a good test-taker, but also by having read lots of books. No internet then, and TV went off at 10:00 pm...
If I'd been required to sit through two years of calculus (as it is presented in The Tomes), and then any of the pedantic [sic] versions of undergrad math, I don't think I would have seriously imagined that I could be a mathematician. More importantly, I would not have wanted to be a party to such grim, oppressive stuff. A substantial part of my luck was to be a good test-taker... which, let's confess, is not really much related to real mathematics. :)
In other words, in the U.S., undergrad grades, GRE, and that kind of thing are not a sample that I find/have-found most useful for imagining U.S. students' future success. Comparisons to students from abroad are difficult.
(About bias: If the question is really about kids in the U.S. with various ethnic backgrounds, then, no, there's no excuse. So, does "Asian" mean "someone literally from Asia, who's gone to school in Asia", or does it mean "a U.S. kid of Asian ancestry"?)