To be fair, the technical reason is that the modern web (circa last 3-5 years or so?) has mostly switched over to demand scaling, and this is why hug-of-death situations are comparatively rare now.
If you have to be super-economical, and even setting an upper spending limit is uncomfortable, set your site up to ping you when/if it's falling apart so you can pull the trigger on ordering more cores yourself. Maybe put a super easy "order more cores" button in the ping, make 2:30AM disasters less terrible.
Of course, technical reasons aside, my initial unimpressedness (or, I must admit, indignation - ha) was directed at whoever this group's ISP is. Clearly the ISP is not providing enough [demand] bandwidth.
[Edit: After reading some comments below I think I should insert a preface/prefix here and say that it's entirely possible the website itself is misconfigured. I think I got sufficiently indignant that I didn't factor this in, and this was definitely an oversight. I don't know the situation, and it's unfair to squarely blame the ISP. Original message continues unedited...]
So, FWIW, this group is using mediatemple. (Found by a simple google of the IP and careful untangling of the results.) ...Well that's a name I've not heard in years. And it made me wonder - do MT have an answer to demand scaling? I'm not able to discern either way from a quick browse of their website (including the "managed wordpress hosting" section). If they don't (and they seem to be stuck in the VPS days), ouch. If they do have demand scaling, then I think it's unprofessional to offer configurations that don't handle situations like this.
In any case, a "504 Gateway timeout" with a tiny "nginx" underneath is going to make people go "hmm, who's this person's ISP" and knock a few points off whatever the answer to that question is. I wouldn't want that happening to me or anyone I build websites for.
[Edit, in the same vein as the edit above: the conclusion above is plausible, but incomplete. Again, it's necessary to consider all angles.]