This argument is fascinating, because you are in effect arguing that people should be rewarded according to their lack of privilege: "we should give these subsidies to students from 'lower-tier schools' rather than students from 'top-tier schools' who are already well on their way to success".
This is the "social justice" argument.
Now let me ask you this:
If you get sick and require surgery, would you rather be treated by the smart, skilled, hard-working doctor who succeeded in everything he tried, including being accepted into a top school?
Or maybe you'd prefer the under-privileged doctor, who never did well academically, and only got into a lower-tier school, but managed to graduate thanks to your social justice subsidy?
Your argument seems to make society more just and equal, but you support it because it is convenient to appear as just person, and you are isolated from the repercussions of your proposal. In reality, your policy will prefer and encourage mediocre doctors over excellent ones, you just hope it won't affect you.
Thankfully, American society is structured in such a way that a top school like NYU can muster the funds required for the subsidy, and thus we will have more excellent doctors, instead of more poor ones.