The problem with weighing other factors is that they are the ones that rich people can trivially fake. An SAT score at least requires the applicant to actually perform good on the test. You can buy tutoring but it won't fix your nepo baby failson's terrible grades in the same way that, say, having a really good essay and a bunch of impressive-looking extracurriculars can.
If your goal is egalitarianism, however, the best response would be to abolish Harvard, not have them implement the Progressive Stack.
The thing is, Harvard education is not measurably higher quality than other schools. You can get a very similar quality of education elsewhere. The real reason why you go to Harvard is specifically for access to the social structures of America's rich and powerful. Every rich family is sending their kids to Harvard specifically because every other rich family does the same, creating a nexus of social capital[0].
Now, if you're a poor or middle class kid that manages to get into Harvard, that's great, you have the ear of a bunch of rich kids all at once. The SATs were the vehicle by which that happened. Intelligence does not accumulate like social capital does, so even an imperfect measure of intelligence makes the pool of accepted applicants more egalitarian. If you want to replace the SATs, you need a selective measure that rich people can't readily optimize for, and by Goodhart's law, such measures are extremely hard to find.
So what would "abolishing" Harvard mean, exactly? I don't mean literally winding down the school's business entities or nationalizing it (mostly), but breaking the social norms that make rich people all cluster in one educational gated community away from everyone else. Harvard won't break them themselves, because it brings in tons of money from rich donors who want to don't-call-it-a-bribe the admissions department. But what if we forced Harvard - and every other university - to share one giant pool of applicants? You'd have a unified admissions system that took every applicant and randomly assigned them to 3 to 5 different schools' applicant pools. Students could then pick from the acceptances they got or try again if they didn't get any. This incredibly invasive change to how university admissions exist would have to be done for the sole purpose of keeping all the rich kids from just picking Harvard.
[0] In general, social capital precedes all other forms of control, including capitalism and politics. See also: The Tyranny of Structurelessness.