Dear Jochen --
Thank you for your kind remarks. I certainly agree that it's possible to say silly things about Gödel's Theorem--perhaps you'll agree even that it tends to attract them.
A few years ago, I spent some time with philosophers associated with the University of New Mexico, an hour's drive (at high-speed across the desert) from Santa Fe. They were constantly employing diagonalization, and importing a huge amount of that mathematics into philosophical arguments. And I spent a lot of time hammering back in a skeptical mode, asking whether the substance of their arguments required this machinery. You can take a look at our reading list here: http://tuvalu.santafe.edu/~simon/undecidables.txt
In the end I was convinced that many of these arguments actually did matter, despite their unfamiliarity and their often strange patterns. And that it was often not too hard to adapt them to problems apparently outside their scope.
For example, consider the vagueness problem you mention. If I get you to attribute probabilities to whether or not a book is meaningful under some sufficiently self-referential criterion of meaningfulness, then I'm already off to the undecidability races. It doesn't matter if I get it wrong from time to time, or if people disagree--as long as there is some content to the notion of "meaning", something above-null and beyond triviality, that I can get at or approximate through debate and discussion, then it's pretty simple to show how all of the disastrous logical machinery still kicks into gear.
Just because a debate exists, in other words, does not imply that there's nothing there to debate--indeed, one can quite easily make the opposite case: that they very persistence of that debate shows that there is some substance there.
By the way, I was just visiting some folks at MIRI in Berkeley yesterday, and (again) I found myself playing the skeptic on these kinds of arguments. One of the things I took away from them, Jessica Taylor in particular, was that these arguments can also provide new kinds of upper (or lower) bounds. The MIRI style seems to be to assume oracles and other forms of hypercomputation and to show that even then, one can't get what one wants.
For that reason, I like your account of incompressibility. Of course, the underlying idea is Komolgorov Complexity, which is itself uncomputable (an earlier draft of the essay had this material, but I was over word-count). So by assuming the existence of a way to compute it, you're actually beginning a MIRI-style argument there. I'm not sure if I want to go the direction you're going. Indeed, there are often meaningful statements that are meaningful just because they can be rephrased or made more succinct--think about how mathematicians find shorter and shorter proofs of a theorem. But perhaps I don't grok it yet.
Finally: I'd agree that this gap is an epistemic one, not an ontological one. That doesn't make it any less real, of course! And because we're epistemic, knowledge-seeking creatures, it has lots of effects in the material world: like causing me to type too long and almost miss my flight, aaa!
Yours,
Simon