I don't know what you mean by "standard situations" though. The "situations" that Bell was considering very specifically dealt with measurements made at a spacelike separation, I don't think he would have said any "proof of quantumness" (in the sense of violating the 'local causality' he was concerned with) can be found in other types of experiments.
I found one of the papers
, but without knowing much about quantum optics, and without being able to read the theoretical proposal that led to this experiment, I don't think I can understand it (I can't even see anywhere in the paper where they compare quantum predictions with some inequality derived from local realism, it seems like all the equations deal exclusively with the quantum predictions and the reader is expected to know which ones violate some Bell inequality). Can you just tell me whether these experiments demonstrating "single photon nonlocality" involve looking at correlations between results at two or more detectors, results which experimenters in the neighborhood of each detector could in principle write down before learning anything about the result at the other detector?
That's true, but a derivation of the claim that violations of the CHSH inequality are incompatible with local realism does require some assumption about measurements made at a spacelike separation.
When you say "nothing of importance about hidden variables", are you denying that experiments of the type envisioned by Bell could definitively rule out any theory of local
hidden variables (where the value of a given local variable cannot be causally influenced by anything outside of its past light cone), or do you just not consider that "important", perhaps because you weren't interested in local hidden variable theories in the first place?
But what does "complex enough" mean? A local theory of the type I described could have arbitrarily complicated values at each point in spacetime (for example the values might be tensors rather than vectors, or any other type of mathematical object), as long as the values at each point were not causally influenced by anything outside the past light cone of that point. Maybe you're talking about a nonlocal theory where the values of the field aren't associated with particular points in spacetime, or where FTL causal influences can occur? I'm not sure I would still call either of those a "classical" field theory, I don't know if that term has a standard definition though.