I love this question OP.
I do think the other students at a prestigious school are the main tangible factor that makes a school “better”. I think it’s entirely likely that you can get fantastic teaching quality at MANY colleges these days, in part because there are many excellent profs who are trying to climb up the ladder too. By no means are all the best profs at HYPSM. Last year, tiny Kenyon College in Ohio had eight profs in the top 300, while Harvard only had four.
As far as name recognition, well, you are right. You will not find many in CA who are aware of WUSTL, but as these things go, the people who need to know, know. But you want to stay in CA. I personally do not think a person’s future outcomes are determined by where they attend school, but a lot depends on the person. Are you a “grow where you are planted” type, or a “will grow if planted in the right place” type? Personally, if I planned on staying in CA (I am from SoCal and am there often) I am not so sure the extra money for the prestige would result in a good ROI. There’s a good story about two students from Indiana or similar. Super Star kid got into Yale, other kid went to State U. Fast forward to several years later, State U kid walks into work and Yale kid is there. They are earning the same money. Prestige doesn’t necessarily mean your life is going to be dramatically different in the future.
I am guessing that you are probably in the Honors college at UA, where you will absolutely find talented and intelligent people. You will probably meet brilliant students who could have got into the best colleges in the country, but they are also at UA honors college because they chose it, or can afford it, or want a big U environment and all that comes with it, and so on. A good friend has a D at Northeastern, honors program. This girl is gifted and got into several tippy top schools. They offered her a ton of money to go to NE and she took it. She LOVES her school, is deliriously happy. She certainly doesn’t regret her choice.
At UA there will also be a lot of bright but maybe not stellar kids and some of them may be your future friends. Plus, a lot of smart kids don’t want to attend a school filled with braniacs, high-achievers, and the like. (My son will be one of them. “Smart people are so annoying.”) I can’t imagine UA doesn’t have good libraries, or lacks the resources necessary for students to succeed. You will probably be a top student at UA, and there is something to be said for being a big fish in a big pond. You will get more opportunities coming your way if you stand out to your profs.
I don’t have an answer to your original question. Really a lot of it is about public perception and hype. Stanford’s dwindling acceptance rate makes the headlines, and the next year, even more people apply. People want what they can’t have, and it’s often totally irrational. Here is something to consider though. U of Arizona has 43,000 students. It’s huge, yet, the school has an 81% freshman retention rate, so the majority of kids are pretty happy there. WUSTL has a higher retention rate of course, but my point is that your experience will be what you make of it. Think about where you would see yourself being happy for four years, and don’t consider the name. That’s probably the school you should go to.