The halting problem is more than that. While the proof constructs a relatively trivial Turing machine that basically puts the toddler's "Nuh-uh!" on solid mathematical ground, that's just the proof that there exists at least one Turing machine for a given halting oracle that can not be predicted by that halting oracle.

To get to where you're complaining about people going, it needs to be paired with Rice's Theorem, as mentioned in the article: https://en.wikipedia.org/wiki/Rice%27s_theorem

It is true that the halting problem itself is perhaps not directly what people seem to think it is, but people turn out to be right anyhow, because Rice's Theorem can be largely summed up as "It can be proven that we can not have nice things", if you'll allow me a bit of mathematician-style humor, which corresponds more-or-less to the way people talk about the Halting Problem.

This also explains why the Goldbach conjecture is in fact directly related to the halting problem and not orthogonal at all. Through Rice's theorem we can extend the halting problem quite easily to the solution to that and a lot of other problems.

By computer science standards, Rice's theorem is relatively basic stuff. I don't recall if the sophomore-level class I learned about this stuff did a full proof of Rice's theorem but I certainly recall several homework problems around applying the idea of the theorem to several specific problems by proving reductions of those problems to the halting problem. I also recall they really weren't particularly difficult problems. It's not hard to reduce that sort of interesting question to the halting problem. Low-level undergrad stuff, stuff down in the very basics of computer science qua computer science suitable for making up several problems in the weekly homework assignment for the intro to (real) computer science course, not complicated abtruse stuff that can't be applied by anyone less than a doctoral candidate with a few months to burn. All sorts of problems are reducible like this.

And that is why the "halting problem" in fact is the reason why we can't have certain nice things we'd really like. For example, the mentioned ability to prove that two functions really are equivalent would have practical utility in a wide variety of places (assuming somewhat reasonable efficiency), like compiler optimizations.

Your intuition is correct in one respect though; there is a sense in which it isn't really "The Halting Problem (TM)". You could take a wide variety of other problems and call them "the" fundamental problem, and reduce the halting problem to those problems. The Halting Problem just turns out to be a particularly concise specific expression of the general problem that is easy to express for Turing Machines. It just makes for nicer proofs than the alternatives. "The" core problem could arguably be something more like Rice's Theorem proved on its own as a fundamental proof without reference to the halting problem. You could call the Halting Problem a historical accident that we hold on to due to a path dependency in how math happened to develop in our history. An alien race might have a reasonably different view on the "fundamental problem", but then they'd have a something fairly similar to Rice's Theorem sitting on top of it that we'd all recognize.