Speaking as a lay person, I think the difference might be that it's specifically about local hidden variables. If two particles are coupled, there's no per-particle hidden variable?
From what I understand, we have some function f that can be evaluated by the measurement and we have another, entangled function g, that can also be evaluated but will yield the opposite of f.
Now instead of assuming that f is truly random and somehow communicates its value instantaneously to g when evaluated, we could also assume that f and g contain a copy of the same pseudo-random number generator and the same seed.
In both cases the interpretation of the model is weird, of course. But I don't see a fundamental difference here that wouldn't allow for hidden variables. It's just that the hidden variables would be have very non-trivial domains.
If I write 0 and 1 on two different pieces of paper, then flip a coin to decide which paper to give you, we have "entangled" unknowns. When I reveal my paper, we instantly know what's on yours. The joint distribution can't be described per-particle, but we don't consider it spooky. So I think there's something more to it.