Edit: Sabine Hossenfelder has explained this much better than I ever can: See for example http://backreaction.blogspot.com/2021/12/does-superdetermini...
Other such assumptions that would allow local hidden variable theories are "a wizard did it" and the rejection of the principle of relativity. All three share that they have no convincing reasoning behind them and that they are largely incompatible with our model of falsifiable science.
The requirement that you can encode Turing machines might sound silly but I really think it's the core of the issue. The ability to make computers implies a certain level of hard-to-control due to the existence of things like the halting problem and pseudo random number generators and cryptographic hash functions.
My understanding is that super determinism is one of those loopholes where it's easy to say its a problem, but actually providung a plausible concrete model where it's a problem is very hard.
I am not aware that such a theory exists, but the Nobel laureate Gerard 't Hooft made some possible first baby steps into this direction:
Gerard 't Hooft; The Cellular Automaton Interpretation of Quantum Mechanics
However, sometimes, people do think about Turing Machines. Here is one paper by Scott Aaronson http://www.scottaaronson.com/papers/ctchalt.pdf
But this doesn't make it wrong. Superdeterminism is a very neat way of resolving physical paradoxes, such as the Wigner's Friend paradox: Any experimenter’s decision to take a measurement of a certain physical phenomenon is predetermined by events, so the experimenter can attain no outcome but the one he does eventually attain, and nothing is in superposition. Neither Heisenberg's uncertainty principle nor Bell's inequality render this condition impossible, as Bell himself was well aware.
The same objection can be stated for the many-worlds interpretation of quantum mechanics:
I rather think the same standard should be applied to both.
It's sort of like constructivist mathematics in that it involves dropping an extremely useful axiom, and then rederives basically all the same results in a more roundabout way.
https://www.quantamagazine.org/why-the-many-worlds-interpret...
"Attempts to explain the appearance of probability within the MWI come down to saying that quantum probabilities are just what quantum mechanics looks like when consciousness is restricted to only one world. [...] What the MWI really denies is the existence of facts at all. It replaces them with an experience of pseudo-facts (we think that this happened, even though that happened too). In so doing, it eliminates any coherent notion of what we can experience, or have experienced, or are experiencing right now. We might reasonably wonder if there is any value — any meaning — in what remains, and whether the sacrifice has been worth it."
That quote is simply wrong when it says that MWI "restricts consciousness to one world". As a simple matter of calculation, you can work out what MWI predicts a complex agent will report as its experience. Here for complex agent I want you to picture a computer program recording events and accumulating statistics, not a person, just because it makes the thinking clearer. Anyways, the prediction is that the recordings will be consistent with a classical type view, for lack of a better word, as opposed to a superposition of experiences. Even if the computer program was in fact run in superposition and interacted with superposed objects.
But this is not surprising, because we already know that all the interpretations of quantum mechanics give the same predictions. So of course many worlds is not going to disagree with collapse type theories on what people say they experience. Because it is mathematically equivalent to the others. Observationally indistinguishable.
Edit: where does the MWI define “complex agent” and “experience”, by the way? I thought that MWI was precisely defined by assuming the postulates of quantum mechanics without including the Born rule, but it seems that there may be more to it!
By Occam's Razor, if there exist multiple theories that are indistinguishable, we should take the one that has minimal assumptions (i.e. no assumption of the existence of some "multiverse" like in the many worlds interpretation).
There's kind of an ontological trick at this point. Using decoherence, self-locating uncertainty, and one or more assumptions about rational decision-making, you can derive what rational credence you might apportion to each of the agent's possible experiences. I feel like this is different from actually predicting the agent's experience.
Those questions do not seem easier to answer than “what’s a measurement”.
Which suddenly sounds a lot like QBism.
Every interpretation of quantum mechanics is utterly untestable, or more precisely, each one is exactly as testable as every other interpretation because they all explain the data equally. That's why they're "interpretations".
This may (or may not) change with work by Gerard t'Hooft and others.