Note: I'm not criticising TeX, but changing fonts away from CM is about the first thing anyone concerned with quality output should be doing.
Aesthetics can't be "cargo culting" -- to assert that it is requires misunderstanding aesthetics, cargo culting, or both.
> It's almost as if adopting it for the web is simply a grasp for mathematical credibility more than any real aesthetic consideration.
Familiarity in context is a real aesthetic consideration.
This means that almost every maths paper they read before was typeset in it, thus to be taken seriously they think they need to be as well. This is cargo culting because it is adopting the trappings of insightful study in the field of mathematics without necessarily doing anything of the sort.
Using the font because it looks respectable isn't cargo cult at all. That's the rational thing to do... That is, once you verify that the effect is real.
If you look on Google Fonts, Playfair Display and Old Standard TT do some of those same things in (I think) also aiming for an old-timey air.
So, I think some of the "air of respectability" comes in part from traits shared with some other fonts, not just from appearing on math papers.
Two, the "feel" and aesthetic aspects of a font mostly do come from memory and associations. nytimes.com doesn't use a serif because it's more readable on the Web or something; they do it because serifs and newspapers go together in most folks' minds. Even the most thought-out arguments about fonts have associations as their bedrock: when Robert Bringhurst laments that a font shows no real axis with its variations in stroke width and therefore fails to evoke calligraphy, the axis he's missing is still important "merely" because of its association with calligraphy. Other than purely technical arguments about readability (which you could dispose of with an A/B test), most opinions about fonts are, one way or another, rooted in associations.
So if folks do appreciate CM partly because of their experience with it I don't see anything illegitimate about that (unless maybe it were utterly unreadable, rather than just quirky). You don't have to use or like the CM fonts, but I don't see how anyone's clearly indisputably wrong for liking them either, any more than with any other font.
CM works pretty well right out of the box (and by default, so if you don't care that's what you get) for mathematics, very few fonts do and most of those are expensive.
What people seem to forget is what a terrible, terrible state of affairs it was that drove Knuth to spend 10 years or whatever on this stuff. Prior to that the options universally sucked.
Even now there really aren't many viable alternatives for typesetting mathematics, and TeX/LaTeX is by far the best of them. But it only comes with one (or two if you count Euler) with full symbol support.
\usepackage[bitstream-charter]{mathdesign}
\usepackage{sourcecodepro}
So all in all it's a bit of a nasty aesthetic trade-off. There are several packages that attempt to create symbol sets to match things like palatino, with varying success. Some are just terrible. And there are a few commercially available ones that do support most if not all symbols, so if it's worth a few hundred dollars to you there is that route.
Nothing has great solution for web, really. We're slowing moving from execrable to ugly but at least it is progress.
I can't really convince anyone on aesthetics, but please look at how the default zoom level on Mobile Safari renders many of the bold characters as having this weird double-line effect:
https://www.dropbox.com/s/enojel6ymuhfnlw/Photo%20Dec%2023%2...
https://www.dropbox.com/s/gkk1aqgduka40gw/Photo%20Dec%2023%2...
Doesn't look any good on my screen either, but it's a print font I'd expect it to become pretty at around 210 dpi.
I think the problem is that none of the TTFs include bold italic, so the browser generates a 'fake bold' of the italic. This time, MobileSafari is doing it oddly. Apparently a workaround is to fake bold yourself with -webkit-text-stroke (icky) and pretty workarounds would be to avoid bold italic or have a separate .ttf for it.