Oh I'm well aware of Bell's theorem; I'm a trained Physicist :)

Bell's theorem rules out theories of local hidden variables. It says nothing about non-local hidden variables, or even something more mundane like determinism (sometimes referred to as "superdeterminism").

In any case, I'm not sure Bell's theorem has much impact on my question: why is one of these things (transfer of information from a coin toss) not a problem, conceptually; whilst the other (entanglement of quantum systems) is a problem, conceptually?

(Personally, I don't find either particularly troubling; just curious to know what the philosophical distinction is, without appeals to "quantum weirdness")

A classical analogue to entanglement is not problematic for causally connected processes and in some cases it's a good model, like with human reasoning or neural networks. QM is a generalisation of bayesian reasoning that can handle non-commuting variables.

Yet entanglement is observed non-locally, even backwards in time or between degrees of freedom that never co-exist. Nobody has been able to create a non-local model that doesn't require fine tuning and the idea goes against the spirit of special relativity.

Thus, I would agree with you, there is no "problem" with entanglement. Except non-locality.