I'm late to the party, so I'm not sure you'll see this. Hope you do. Different people will each have their own contexts and their own concepts of "useful" or "interesting". I'm making assumptions about yours ... apologies if I misrepresent you.

The proof of Gödel's result uses the paradoxical statement "This statement is false", but that's being used to prove this very general result about all systems. So the hunt is then on to find "natural" statements that are True but Unprovable.

But the "unprovable" bit should more completely be stated as "unprovable in a specific axiomatic proof system". If we want to prove that statement S is "True but Unprovable" then we must actually prove that it's true. So if we've proved it's true, what does it mean to say it's unprovable? We just proved it! What's going on?

So let's take a specific example.

Peano Arithmetic (PA)[0] is an axiomatic proof system intended to capture Natural Numbers and their behaviour.

The "Goodstein Sequence" G(m) of a number m is a sequence of natural numbers ... you can find the definition here[1]. It's not hard, but it's longer than I want to reproduce here.

Goodstein's Theorem (GT) says that for every integer m greater than 0, G(m) is eventually zero.

It has been proven that GT cannot be proved in PA, but it can be proved in stronger systems, such as second-order arithmetic.

So the statement of GT is not self-referential, along the lines of "This Statement Is False" sort of thing. It's an actual statement about the behaviour of integers, so it's not a self-referential trick.

Your question now is: What's the point? How is this useful or relevant?

Much of modern (pure) mathematics is chasing things because the mathematicians find them interesting. The vast, vast majority will never, of themselves, be useful by (what I expect are) your standards.

But it was once thought that factoring integers was of no practical use, and only pursued or investigated by cranks. Imaginary Numbers were thought to be bizarre, useless, and dangerous. Non-Euclidean Geometry was thought to be utter nonsense, and held up as part of the "proof" that the fifth postulate was unnecessary and was deducible from the other four. All three of these now form critical components in modern technology.

Even more, to the average person on the street, anything to do with algebra is completely pointless.

For you, Gödel's theorem is completely pointless and useless and probably of no interest at all, but it helps us understand the limitations of formal systems. The techniques that have been developed in the time since it was proved have helped us understand more about what computer verification systems might or might not be able to accomplish.

Of itself, Gödel's theorem might not be of direct, immediate, and practical use, but the work it has inspired has tangentially been useful, and may yet be moreso.

But not for everyone. After all, some people don't care about the Mona Lisa, or Beethoven's Fifth Symphony, or Michaelangelo's David, or the fact that people have walked on the Moon, so why should people care about results in Pure Mathematics?

That's the thing about Pure Mathematics. Sometimes it ends up being useful in ways we never expected.

[0] https://en.wikipedia.org/wiki/Peano_axioms

[1] https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_...