This sense of how certain or uncertain a probability is may have no place in a perfect Bayesian reasoner but I think it is meaningful information to consider as a human making decisions under uncertainty.

I don't think the key issue is the imperfect Bayesianism of humans. I suppose that the discussed certainty of a probability has a lot to do with its dependence on priors - the more sensitive the probability is to change in priors we find arbitrary, the less certain it feels. Priors themselves feel most uncertain, while probabilities obtained from evidence-based calculations, especially those quasi-frequentist probabilities, as P(heads in next flip), depend on many priors and change in any single prior doesn't move them too far. Perfect Bayesians may not have the feeling, but still have priors.