There's a section in Yes, Prime Minister where Humphrey Appleby makes some comment about preserving England's great universities. Then pauses a beat and adds "both of them." (referring, of course, to Cambridge and Oxford.) It's obvious there are other very good universities in the UK and I don't doubt the LSE has programs that surpass the others. But I spent a couple months on a research project at LSE (and even delivered a few lectures) but most people hearing I was a guest lecturer there were like "meh. whatever." (Oddly, the guy I knew from Cambridge was "oh! they have some very good programs there.")
So... yes... despite consistently ranking high on surveys, Durham and LSE are not "sexy" in the way Cambridge and Oxford are.
I know - I went to Oxford instead of LSE for my masters degree exactly because the reputation of the latter is not there outside a narrow subset of finance/econ professionals in the UK. Even though the LSE programme was older and more established and cheaper.
The same is true in the US as well: The University of Chicago and University of Pennsylvania are probably world beating for economics and finance research respectively -- but I would still recommend people to go to Harvard/Stanford for an economics-focused undergrad or finance MSc or MBA if they have the choice, due to the name recognition and the network...
Sure they'll never have quite the same cachet, but it's the same anywhere - UCLA/Stanford/Yale are extremely respected but nevertheless not Harvard or MIT. No doubt someone more familiar would say not all IITs are equal, but Bombay, Bengaluru & friends lead the pack. &c.
Yale is part of the Ivy League and was founded ~65 years after Harvard. Also ranking #1-#2 for producing US presidents, Harvard-Yale is probably a somewhat better US university analog to Oxford-Cambridge.
Stanford is well-regarded and may be a solid competitor to Harvard in a number of ways (#1 in Turing awards, #2 in VC-backed startups behind Berkeley, etc.) but it was founded 250 years later (considered a long time in the US) and has a smaller endowment (4th place, behind Harvard, UT, and Yale.) It has also only produced one US President: Herbert Hoover.
MIT is a top tier school though more focused on technology (it's in the name). MIT has a business school (Sloan), but a Harvard/Yale/Stanford will include a business school, law school, med school (etc.) and a range of well-regarded programs in humanities and social sciences in addition to science and engineering.
I may have an overly STEM-centric view, but I don't think Yale has anything like the reputation of MIT internationally. I'm only really aware of it from Americans (in Hollywood, newsletters, etc.) being impressed by lawyers' and MBAs' credentials.
Anyway, I don't think the specifics really matter, point is it's not unusual to have a bunch of extremely good universities and then a handful or fewer that are for whatever reason the first-to-mind 'best' ones.
This is likely; international vs. US also probably makes a difference, as three of the last six US presidents (by person, not year) were Yale alumni (no MIT alumni have yet become president, but I think it's a good idea!)
And five are Ivy League grads: Obama from Harvard Law School, and Trump with an undergrad degree from Wharton/U. Penn. (Biden being somewhat of an outlier, having attended U. Delaware and Syracuse University.)
So... yes... despite consistently ranking high on surveys, Durham and LSE are not "sexy" in the way Cambridge and Oxford are.