I take some issue with the idea that we can simply just rely on another equilibrium refinement and say "no big deal".

It seems a bit silly to observe some behavior in a game and say "See that's a Nash Equilibrium, so game theory works", and then to turn around and observe some non-NE behavior and say "well in this game, BNE is clearly the right model, so game theory still works". And then yet again to observe some more behavior that conflicts with the theory (or to get rid of silly equilibrium) and say "ah, now we simply use perfect Bayes" or trembling hand equilibrium, or actually we were totally using correlated equilibrium this whole time.

In any case, it feels weird that a theory should behave like the "No true Scotsman" fallacy. We can always get the equilibrium by simply redefining what we mean by equilibrium.