Just because a school has more choices in picking students does not guarantee the wisdom of its pick. Steve Jobs would have been rejected by all Ivies today. But since the same AO has been picking students over many many years, we do see some pattern that reflects the AO’s taste in part.

I will answer this based solely on my prejudice, for what its worth. Take it at your own peril. (Don’t give me which school is not Ivy stuff - I know.)

1. Harvard: Model human beings with presentable stats and characters (yes, they are genuinely nice), with no evidence of glaring mental disease (see Yale and Princeton for comparison). Most balanced Ivy. The only Ivy with human mascot; all others are beasts. Earth’s answer to alien invasions. Must be, and look, strong across the board, but more importantly, must have no weakness, nothing controversial, especially on paper. Certainly a fox type, not a hedghog type. (“A fox knows many things, but a hedgehog one important thing.”) Righteous and virtuous. Downside: Naive, bookish, unresourceful, unresilient, weak mental, carrying self-congratulatory smile. Often fall preys to determined and/or scheming underdogs. Diploma likely to end up being life’s greatest achievement. Real life is very different from school. Too risk-averse to try something that may be “unworthy” of alma mater; looking respectible becomes a burden after a while. Unable to reject expectation of others. Haunted by the self-question: “What is ‘me minus Harvard’ worth?” Bullied abruptly by bosses: “Let’s test how smart a Harvard guy is.” Bullied abruptly by spouses: “[You don’t even know how to turn off the dang faucet] - tell me, did you really go to Harvard?”
2. Yale: Creative, Passionate Artists with ADHD. Most artistic Ivy. Possess one big thing, lack others and proud of it. Ivy with greatest number of mathematically challenged - you can still succeed in life without understanding calculus. Certainly the hedgehog type (“All I need is making one big hole”) - an outlier with a nuclear punch. Flexible, witty, resourceful, irreverant, pungent, unique. Capable of counter-intuitive, original thinking. Social and gregarious like wolves (in contrast to the tigers that come below) and carry “secret club” antic to life after college. Think they can beat nerdy Harvard any time. Think they cannot beat Princeton, but rarely think of Princeton anyway. Downside: George Bush, George (another) Bush. Can be too creative for own good. Superficial and/or scheming (Many early CIA members were Yalies). Lazy underachievers - and proud of it.
3. Princeton: Rigorously Trained Tripartite Aristocrats (Gentleman+Scholar+Athlete) - with OCD. Most analytic Ivy. No weakness in reality (ie, not just on paper). Most hard working among HYP. Superachievers and fierce competitors. Mathematically comfortable. Motto: Only paranoids survive. Regularly beat both Harvard and Yale in almost everything. Prefer working alone, like tigers (why collaborate when perfection is attainable as solo?). Downside: Robot. Can be too perfect for own good. Serious, ambitious, studious, logical, wicked smart. Brutally efficient like Amazon dot com, lacking idealistic, romantic, human touch.
4. Stanford: Think Duck floating on a lake. But it aint your ordinary, garden-variety duck. For one, it is dressed in the most exclusive designer clothing everyone knows of. Further, its demeanor perfectly complements what it is wearing: With most graceful movements, it tries to look beyond the hills and forests afar, as if searching for long-lost ideals and romances. However, to your surprise, just underneath the water surface, you find its muscular legs are moving feverishly and with brutal efficiency. To complete your surprise, on closer examination, you find something is attached to its feet: it is wearing motor-driven, turbo-charged, state-of-the-art fins! And all of this is happening while the whole lake is glowing with warm sunshine. There you have the essence of Stanford put together: Harvard in packaging (clothing), Yale in spirit (duck’s top half), Princeton in reality (duck’s bottom half), MIT a bonus gear thrown in (snorkeling fins), under the gorgeous sun. Yes, S=(H+Y+P+M)*💥Sometimes it even works.
5. MIT: Too busy thinking anything other than weekly problem sets.
6. Caltech: Better than MIT. QED.