it's the nature of every age to think its problems unique, and the role of history to point out they aren't. Taking a historical stance towards the problems of undecidability (etc.) is thus, I think, a necessary and welcome addition to this essay contest.
Furthermore, there tends to be a kind of intellectual chauvinism directed towards past generations---they back then were just the unenlightened rubes that didn't yet have the benefit of our advanced scientific understanding. So, what could they conceivably have to tell us that could help with our present-day issues?
I read your essay as, in part, questioning these intuitions. The example of heliocentric versus geocentric models is a striking one---data itself does not adjudicate between these views; what does, then, is extra-theoretical constraints, such as e. g. a principle of parsimony. But in what sense is swapping one such constraint for another tantamount to 'scientific progress'?
I'm reminded of the oft-related exchange between Ludwig Wittgenstein and Elizabeth Anscombe---as relayed by her: "He once greeted me with the question: 'Why do people say that it was natural to think that the sun went round the earth rather than that the earth turned on its axis? I replied: 'I suppose, because it looked as if the sun went round the earth.' 'Well,' he asked, 'what would it have looked like if it had聽looked聽as if the earth turned on its axis?'"
You also touch on Hume's problem of induction. This has, in fact, a closer tie to the present contest's theme, in that a straightforward algorithmic formulation of induction, incorporating a principle of parsimony in the form that more simple hypotheses should be given higher weight, ends up being formally uncomputable---this is the theory of Solomonoff induction. Thus, if one is prepared to accept that an algorithmic formalization with a requirement of parsimony is a sufficient characterization of the notion of induction (and one may well not be), then it turns out Hume's problem is exactly equivalent to the unsolvability of the halting problem. Thus, given this formulation, it seems that your notion that "the problem of induction is a point of contact between different notions of undecidability" might be pretty spot on.
You propose an interesting interface between science and philosophy, with the open-ended questions of the latter being brought under a certain formalizable framework by the former. If this is true, then it seems a straightforward corollary is that we'll also never run out of either: for it will always be possible, given a question having transitioned from philosophy to science in this way, to ask the meta-question of wether this transition was truly appropriate---whether, in your words, the ampliative assumptions needed to give form to a given question are actually apt.
It seems to me that this process can also fail badly---for instance, take the question 'Why is there something rather than nothing?', that some physicists take to be answered by phenomena such as vacuum fluctuations, or quantum tunneling between different vacuum states. To me, this 'solves' the question by essentially substituting a radically different one---one about how a certain kind of state matching what we observe today can emerge from a different kind of state that can be thought of as a vacuum state. I think this essentially loses the point of the question---and moreover, I'm not convinced that it's not systematically the case that certain questions don't really survive the spawning of 'daughter' questions to be attacked by the sciences. Perhaps you gain an answer, but only at the expense of the original question.
Regarding the completeness of quantum mechanics, I have a few things to say in my own entry into this contest that might be of interest to you.
Finally, as for the question of consciousness, I have recently put forward a proposal---'The Abstraction/Representation Account of Computation and Subjective Experience', Minds & Machines (2020), https://doi.org/10.1007/s11023-020-09522-x---to the effect that a 'consistent set of principles or hypotheses, possibly new ones, which would allow one to conclude the existence of consciousness (or lack thereof) in a given system' does not exist; at least not in the sense that there could be a chain of rational deductions, or equivalently, a computation, that decides the presence of consciousness within a physical system. Still, I view this as a thoroughly physicalist position---there's a distinction to be made between physics, the science, and physical stuff. This, I believe, is the origin of the hardness of the hard problem.
Anyway, as you can see from the length of my comment (brevity never has been my strong suit), I found much of interest in your article, and lots of food for thought. I think the perspective you present on this contest's topic is an original and necessary one, and therefore hope your article will do well in it.