For more on why “non-informative” priors, particularly diffuse priors on logit-scale parameters, are dangerous, check out this preprint from Subhash Lele, with a focus on problems in population estimation. Lele isn’t a big fan of Bayesian inference in general, but the particular criticisms in this manuscript need not be read as general criticisms of Bayesian inference but rather as criticisms of poorly considered priors that are assumed to be of little practical significance as long as they are “vague”.
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This is all a great discussion on the modeling challenges, but one thing to keep in mind is that Bayes is most powerful when considering the full posterior and not just any point estimate like the mean. The posterior distribution quantifies not just one model configuration consistent with the data but all of them! Even in the cases where N is small and the posterior mean drifts from the true parameter value, the bulk of posterior contains the true value reasonably well. Consequently when using the full posterior, for example reporting uncertainty intervals instead of just the mean point estimate, you would be unlikely to be mislead.
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