For someone not from the US: From the second point I understand that the SAT is not the only thing used to decide who gets in? So, it's not "we have 100 slots this year, the 100 best get in"? What else is used?
Harvard probably wants a mix of people who have a history of going to Harvard, elites, with high performers and others. If it's only elites, then the academic quality goes down. Only best academic performers, elites will go to another school to call home, etc.
If you have the resources to review applicants individually, sometimes there are people who clearly stand out for reasons unrelated to test scores. I expect that in most cases, if they've taken tests, they have good scores, and it can make sense to generally have a threshold, but if you simply rank by top scores they are likely not the absolute best, as they would have been spending more time and attention on things other than studying for them.
Simultaneously, I've also heard that it's easier to get your name on a paper as a high schooler than as an undergrad working in the same lab. Some of this is because professors are incentivized to put random high schoolers on their papers -- when applying for grants, they're often asked what K-12 outreach they've done (https://new.nsf.gov/funding/learn/broader-impacts). "Look at this paper where there's a random high schooler on it" checks the box if a bit inappropriately. Incentives at work!
But in my lab at least, we did have the odd high schooler/undergrad, whom I had a part in mentoring. For the high schoolers, my PI at least privately admitted it was because their parents had "means" and it was about at least the potential for grant money.
I expect very few schools just mechanically admit based on SATs.
Harvard might not have this problem, but some, even some good, schools have to put some thought into if the student is actually going to accept the offer as well. Some states require a certain percent of students to be from the state in state funded schools.
I think most schools would claim it is that, they just reject that testing is the end all be all of who is the best. These schools are looking for the next generation of leaders, theres a lot that goes into being a leader that isnt counted on standardized tests. Standardized tests really just tell you who has the highest IQ. Useful, but it misses lots of other attributes.
If a university's mission goes beyond just collecting already-top-performing students from the best (usually wealthiest) school districts, you can't just do a straight rank on a single signal. There's a lot of variance in educational opportunity across the US, ranging from highly selective specialty public high schools [1], to places where they can only afford 4 days of schooling a week [2].
[1] https://en.wikipedia.org/wiki/Thomas_Jefferson_High_School_f...
[2] https://stateimpact.npr.org/oklahoma/2022/10/27/oklahoma-law...