I guess we should all be glad that the Internet was not designed by Pratchett. If it was possible to send infinitely repeating messages like he describes, the whole Internet would soon be completely bogged down with useless overhead packages filled with spam and ads. At least the fanboys who implement this nonsense header on their servers are only wasting the bandwidth of themselves and their visitors.
Yes, if the Internet worked on the clacks protocol, it wouldn't work in real life. But neither would cameras if they relied on small creatures living in them and painting the photographs very quickly. It's not even that Discworld is horribly naive, it's more that it served as a whitewashed canvas to allow him to create narratives by introducing elements of the real world individually and give the reader the opportunity to draw parallels from that.
The clacks "GNU" hack is a perfect example of the kind of things you'd see in the early Internet days before commercial websites were really a thing. Remember that for a long time there was a dream of the "semantic web" which turned out to be similarly exploitable and fell apart when commercial interests became relevant (giving birth to what we now call SEO).
The X-Clacks-Overhead header is clearly not intended to behave like the clacks in the story. Nobody is naive enough to believe that this protocol would work (just as the semantic web didn't). It's entirely a tribute, a form for fans to (at least temporarily) "immortalize" someone who had a great influence on them. And unlike the twenty different analytics and ad scripts you now find on many websites it not only wastes hardly any space but also doesn't meaningfully inconveniece users (not any more than forgetting to turn of the server and x-powered-by headers at least).