It's worth noting that this is not a case of "elites" taking out an unfairly large share out of a communal resource, it's the case of the elites choosing how much of their
private resource they want to share with the rest of the society. It (e.g. Harvard) does not belong to all citizens/taxpayers, it belongs to the elites, and they can get as much 'leg up' as they choose.
It would be perfectly cromulent for Harvard to choose to admit only legacy students, and it could be a reasonable choice for them to set the tuition fee to a million dollars per year, with no scholarships whatsoever, and admit only rich students. They're not part of a government-funded education system with a goal of educating the whole society, they are a private school who can educate whomever they want - they can go back to teaching only Puritan clergy as they once did, if they so wish.
In essence, all the (quite welcoming!) financial aid and scholarships they're offering for talented students is freely given charity, funded by the fees of the rich students and gifts of alumni - and here we are complaining that they're not doing enough charity and not doing it in the right direction, forgetting that they might as well stop doing it at all if they wish.
They are essentially controlled by their alumni, who elect the overseers, so if the alumni collectively decide that it's good to invite talented-but-not-rich students with scholarships, because they believe it's best for their school and the children of alumni who also will be joining the school as legacy admissions, then that's their right to choose so. It's "their club" and they get to choose which non-super-privileged (if any!) they want to invite to their club.
This is in deep contrast with most other countries, where the elite universities tend to be funded and run by the government, who then gets to set policy to achieve relevant social goals, but in USA the top universities are private and their policy is controlled by the super privileged, not anyone whom the general public can elect.