SUNS OUT GUNS OUT

Wittgenstein found an elegant solution to this kind of paradox: "This is obvious bullshit. Next."
It was a response to philosophers who were trying to come up with a robust axiomatic system for explaining math. Russell and Whitehead's Principia Mathematica attempted to formalize everything in math, and Goedel proved it was impossible.
So yes, it's a bit of a circlejerk, but it was a necessary one to break up another circlejerk.
While what you write is not too far from the conclusion of Gödel's proof (he proves you can construct a statement which is equivalent to "this statement is not provable"), the point of Gödel's theorem is that you require a minuscule amount of language (just enough to work with numbers) to do this.
English is very complicated and not a very formal language, so it's less surprising that you can come up with unprovable statements like yours. Building such a statement in a language that can barely talk about arithmetic is not obvious at all in my opinion. People had already spent a good amount of time choosing a system of axioms that made certain paradoxes impossible to write (for example Russel's paradox, "does the set of all sets that don't contain themselves contain itself", can be written in english, but not in ZFC, the most commonly used axiomatic system in math), and they thought they reached a point where they had fixed all of these paradoxical statements, but Gödel proved not only that they were wrong, but that their goal of a perfect set of axioms where everything could be proven or disproven was impossible to reach.
Also, there are unprovable statements that don't look anything like yours, like the continuum hypothesis: "there is no infinity that is larger that the number of natural numbers, but smaller that the number of real numbers". This a perfectly reasonable statement, not only in english but also in ZFC, which looks like something we should be able to either prove or disprove, but in fact we can't do either. If you want you can add it (or it's opposite) to the axioms of ZFC, getting a new set of axioms, and you shouldn't find any inconsistencies. After Gödel's proof people started asking themselves "is this thing that I'm trying to prove even provable?", which I don't think happend very often before.
By the way, this ability to talk about arithmetic is fundamental to the proof: euclidean geometry can't do that, so Gödel' theorem doesn't apply, and it turns out that it's both consistent and complete.

Huh, framed like that, that seems like a wild statement considering he later went on to formulate his ontological "proof", which attempts to prove God's existence without relying on axioms (and in my not-so-humble opinion fails to do so, because it assumes "good" and "evil" to exist).

But what I'm reading about his incompleteness theorems, it does seem to be a rather specific maths thing, so would've been a big leap to then be discouraged in general from trying to do proofs without axioms.

I don't know much about this, but I can't help but think that "complete" and "consistent" are doing a lot more work in that sentence than my current understanding of the terms would lead me to believe.

I am sure there was a typo, it's Gödel's incompleteness theorem which proves that consistent systems are incomplete.
Consistency means likely what you expect: it's that you cannot reach contradiction from very axioms.
The result is insane in my opinion, it means any sensible math system with basic arithmetic has a proposition that you cannot prove. AND you cannot also prove that the system is contradiction-free.
It is completionist's worst nightmare.
- The result is insane in my opinion, it means any sensible math system with basic arithmetic has a proposition that you cannot prove.
  Stated more precisely, it has true propositions that you cannot prove to be true. Obviously it has false propositions that can't be proven, too, but that's not interesting.