Dear Professor Hossenfelder,
Thank you for this highly original contribution to this contest, which reminded me of Wittgenstein's last philosophical thoughts, collected in "On Certainty", to the effect that most discussions take place within a 'container', but some discuss the container (which he calls the set of hinge propositions) itself. Your essay seems to belong to the latter category, since you challenge, quite rightfully, the whole subject of this contest.
I just don't think that your section 4 soes real justice to the reduction-emergence debate (you may deliberately avoid the latter term), which is so much wider than the context of effective QFT. For example, the emergence of the classical world from quantum theory (if only in theory) relies on the role of small perturbations, whose importance grows near the classical limit and/or in large systems and which make non-classical states collapse; this has nothing to do with RG or effective QFT arguments, and there are many other examples of emergence like that - if emergence seems to fail, you just have to look harder.
Apart from this, and the tiny criticism than in paraphrasing Gödel's Theorem you seem to conflate provability (which his theorems are about) with truth (which notion unnecessarily complicates matters, although admittedly your rendition is seen very often), I really enjoyed your essay, which will give me food for thought form some time to come. A relevant context which I like is "Earman's Principle" from the philosophy of science, which states that "`While idealizations are useful and, perhaps, even essential to progress in physics, a sound principle of interpretation would seem to be that no effect can be counted as a genuine physical effect if it disappears when the idealizations are removed.". Your claim seems to be that the three theorems about the "un's", which from the point of view of physics are indeed idealizations, do not satisfy this principle, but this remains to be seen. With the idealized assumptions weakened, these theorems may still make weaker but valid and interesting claims about the real world (or at least about theories thereof). It is often very challenging to find such weaker but de-idealized versions of mathematical theorems, but they typically exists, for example, in approximations to the law of large numbers. An example relevant to your essay in spontaneous symmetry breaking, which according to official idealized theory can only occur in infinite systems (which would be very puzzling since it is seen in many finite materials), but which in fact is foreshadowed already in finite systems, once treated correctly. I certainly believe that this also applies to Chaitin's Incompleteness Theorem which you mention (and I implicitly use this in my own contribution to this contest) and I presume there should also be weaker version's of Gödel's Theorems to the same effect.
Best wishes, Klaas Landsman