Disagree. Why are those CMS so god-awful? What're the incentives that they're designed to achieve? Badness in tooling is usually a symptom, not a cause.
It's a byproduct of a one-size-fits-all-content design for a CMS that all of their output looks like the same drivel. In contrast, if a designer had the opportunity to hand-design each content page distinctly, such a site would look markedly better.
CMSes are not built to produce beautiful design, but to expedite content publication into a designed-once template, so it's no shock that their output typically exemplifies bottom-barrel design.
"It's a byproduct of a one-size-fits-all-content design for a CMS that all of their output looks like the same drivel."
I think this is a really important point, and want to further expand on it. The New York Times was cited as having a bad website. However, on occasion, when they put their mind to it, they produce a truly beautiful page with some fantastic visualization or something. But it's a one-off; you can't produce a beautiful page for every story, so you get the lowest-common-denominator page for most stories.
A literal paper magazine is a concrete artifact, and a set of designers can be tasked with making each page perfect. A web page on these sites is dynamic and may literally not serve the same page twice. If, as a designer, you don't internalize the idea that these are two extraordinarily different domains, despite the superficial visual similarities of the final product, then you're going to have a hard time accurately diagnosing the core problem. And I do believe many designers make this error on one level or another, when I see these complaints.
In Matthew Butterick's defense, I do think he did implicitly make one very concrete suggestion for how those sites can improve, though, at least in terms of design: Get rid of the ads. It's hard, probably to the point of impossible, to create a coherent design for a website when you are selling to the highest bidder the right to put a bright yellow and orange banner with a poorly-photographed item described in a random non-anti-aliased font unaligned with anything else on your page in the viewer's highest-priority visual space, and a different such thing for each page load. Against an uphill battle like that, is it really any wonder that the web designers of the world have collectively given up?
CMSes are not built to produce beautiful design, but to expedite content publication into a designed-once template, so it's no shock that their output typically exemplifies bottom-barrel design.