Incompleteness means that maybe it's actually impossible to prove or disprove the Riemann hypothesis, or whether P=NP. I think that's a big deal. Maybe the fact that there are meaningful results shown to be unprovable (like axiom of choice) is more satisfying?

with a rigorous, needlessly convoluted proof.

Again, Goedel's theorem was in direct response to Russell and Whitehead spending literally decades trying to axiomize mathematics. Russell's proof that 1+1=2 was 300 pages long. It was non-trivial to disprove the idea that with enough formality and rigor all of mathematics could be defined and proven. Instead of the back and forth that had already taken place (Russell proposes an axiomatic system, critics show an error or incompleteness in it, Russell comes back and adds some more painstaking formality, critics come back and do it again), Goedel came along and smashed the whole thing by definitively proving that there's nothing Russell can do to revive the major project he had been working on (which had previously hit a major setback when Russell himself proved Russell's paradox).

how about:
x = 2
2x = 3,000
omg! they’re inconsistent!

You didn't define x, the equals sign, the digit 2, 3, or 0, or the convention that a real constant in front of a variable implies multiplication, or define a number base we're working in. So that statement proves nothing in itself.

And no matter how many examples of incomplete or contradictory systems you come up with, you haven't proven that all systems are either incomplete or contradictory. No matter how many times you bring out a new white swan, you haven't actually proven that all swans are white.

And formal logic and set theory may have seemed like masturbatory discipline with limited practical use, but it also laid the foundation for Alan Turing and what would become computer science, which indisputably turned into useful academic disciplines that changed the world.