And if so, is the reason why no one has gone for the more bespoke route is because no one who programs is in charge of these companies, and thus does not even know that they could benefit from such an approach?
The only parallel I have some experience is in the publishing industry. No one thinks that the technical details of the CMS is important, as long as stories are visible on the Internet. But if a clunky CMS requires you to hire 10 to 20 people just to move pieces around on the frontpage (this is on top of the backend that automatically adds stories to the system and to their proper categories), then that is not a trivial cost.
And I'm sure The Verge, with its custom CMS (in Rails, it seems) has some clunkiness to deal with, but they've been able to produce varied and interesting formats for their stories on a much more everyday basis than even the New York Times (I'm not counting interactive graphics/multimedia, I mean the general story form).