For example, if aliens claim their machine solves the halting problem, we could test it on millions of inputs whose halting/not-halting behaviour we already know; but even if it works for all of them, there's no way to know that it works for all inputs. For all we know, it might be a huge lookup table which happens to cover all of those inputs we tried.
> if our universe is undecidable
My point is, there would be no way to empirically test this; and therefore, it would make no observable difference, there would be no way to exploit/utilise such effects, etc.
In essence: there's no way to tell the difference between a real halting oracle (which would imply an undecidable universe), versus a computable approximation which just-so-happens to be more powerful/sophisticated than the approximations we compare it against.
Sure, we can prove that some abstract systems are undecidable and that others aren't. Yet that distinction is inherently unfalsifiable, and hence physically "useless".