@Penn95 I have no idea why UChicago’s Rhodes numbers aren’t great. Level of interest might have something to do with it.
To compare it to other schools we’d need to look at (to start…) the percentage of applicants who become Rhodes scholars – not just the number who obtain the honor.
Assuming relatively equally bright and driven students (always a big assumption, because individuals are individuals…) at the Ivies and Ivy equivalents, if School A had 100 apply and 3 make the cut while School B saw 50 apply and 2 become Rhodesians (ahh, late-night mirth, what a lovely thing you are!), I’d say school B was slightly more efficient at producing Rhodes scholars, while School A was more effective. Pot-banging and chest-thumping ensue between those who count on their fingers and those who analyze.
Maybe School A has twice as many students, accounting for the 100% greater number of apps; but if it has fewer than twice as many, maybe they spend more time pushing students to strive for such honors and opportunities. Or maybe they simply attract kids more likely to strive for such things.
If GPA is an important factor, we need to look at average GPAs at schools, both overall and taking into account how popular each major is at each school, and average GPAs by major. If UChicago has a low Rhodes admit %, maybe grade deflation is a culprit.
But really, I have no idea where UChicago falls in terms of Rhodes efficiency or effectiveness – for all I know, lots of them pursue it and few succeed – but without the simplest analyses and comparisons we can’t really know if they’re just ineffective… or also inefficient.
In junior high, I knew that if I shot enough spitwads at the ceiling, eventually one would stick. Drive and perseverance (and multiple attempts…) produce results. hehe
Regarding the intellectual level of a campus, it is crude and probably not precise to say, maybe, but I think there might be a little bit to this: that the greater the percentage of Social Science and Humanities majors there are, the higher the rate of intellectualism. Those kids write tons of papers and engage in classroom debate probably moreso than the average STEM major. But wait… that only looks at majors that tend to encourage debate.
Can a campus – not just specific majors – act as a magnet for those who love intellectual pursuits? A place where even the comp scientists can knock your block off with their voracity for reading, research, analysis, extrapolation, dissemination and debate? Maybe that’s what we’re after here: yes, schools with a lot of non-pre-professional majors will tend to have a more intellectual vibe… but maybe the truly intellectual kingpins are those schools where the Math kids are arguing over coffee about something in their Lit class.
Maybe having a core pushes that agenda a bit: you can’t dodge the great books or the big thinkers. You are made to care. You will be… intellectualized. 
I figure these three schools can all be very intellectual for whomever seeks that vibe. Because it lacks a core, it might be a little bit easier at Princeton to avoid intellectualism, but then there’s the senior thesis, which must be well researched, structured, reasoned/argued, and written – intellectual with a capital I.
Three very special schools with three very different vibes, but all are great at educating those who wish to learn.