Dear Prof. Landsmann,
this is a very exciting essay! I have only given it a first pass, but as far as I understand, you propose to extend the scope of Bell's theorem from the statistics of ensembles of measurement outcomes to the characteristics of individual outcome strings, thus uncovering the incompatibility of quantum mechanics with (a certain notion of) determinism.
I think this is a highly original way to think about these issues; certainly, most treatments never leave the level of statistical analysis, but of course, the statistical properties of an ensemble don't suffice to fix those of its members. I'm reminded of the old joke: the average human has one testicle and one breast, features which a 'theory' of beings that have one testicle and one breast each may well replicate; but that theory would fail badly at reproducing the characteristics of humans on an individual basis.
Again, if I understand you correctly, your main argument is that the deterministic replacements of quantum mechanics fail to replicate the typicality of individual outcome strings, while meeting the requirements posed by the Born rule. That outcome sequences of quantum mechanics must be Kolmogorov random has been argued before, in various ways---Yurtsever has argued that computable pseudorandomness would lead to exploitable signalling behavior (https://arxiv.org/abs/quant-ph/9806059), echoed by Bendersky et al., who explicitly prove that non-signalling deterministic models must be noncomputable, if they are to recapitulate the predictions of quantum mechanics (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130401). Likewise, Calude and Svozil have argued for non-computability from a generalization of the Kochen-Specker theorem (https://arxiv.org/abs/quant-ph/0611029). (And of course, there are my own efforts, see https://link.springer.com/article/10.1007/s10701-018-0221-9, which also makes use of Chaitin's incompleteness theorem, and my contribution to this contest.)
Thus, the randomness of any quantum outcome sequence can't be produced by any effective means, and hence, any deterministic theory must either fail to reproduce these outcome sequences, or otherwise incorporate this randomness by fiat (as in the Bohmian equilibrium hypothesis), which renders its origin essentially mysterious.
It seems to me that at the heart of this is really the observation that you can write any noncomputable function as a finite algorithm that has access to an infinite reservoir of (algorithmic) randomness. In this way, Bohmian mechanics can play the role of the algorithmic part, which has to be augmented by an infinite random string in order to replicate the quantum predictions.
There is, however, another way that's quite popular at present---you can also just compute any sequence whatsoever in parallel, by an 'interleaving' algorithm that just outputs all possible bit strings, running forever. A measure-1 subset of the strings produced in this way will by typical, but the overall operation is, of course, quite deterministic. This is basically the sense in which the many worlds interpretation is deterministic: if we just look at any given bitstring as a single 'world', then in general one would expect to find oneself in a world that's algorithmically random.
Another observation of yours I also found highly interesting, namely, that in principle the non-signalling nature of quantum mechanics should be considered as a statistical notion, like the second law of thermodynamics. In the limit of infinitely long strings, the non-signalling principle will hold with probability 1, but for finite lengths, deviations may be possible. However, one probably couldn't get any 'useful' violations of non-signalling out of this, because one could probably not certify these sorts of violations (although perhaps one could provide a bound in such a way that Alice and Bob will agree on a certain message with slightly higher probability than merely by guessing, with that probability going to the guessing probability with the length of the message).
Anyway, thanks for this interesting contribution. I wish you the best of luck in this contest!