Yes, I know, but, asymptotically, the quantiles of the posterior will define correct confidence intervals under the frequentist interpretation (see An MCMC approach to classical estimation - ScienceDirect, theorem 3, for example).
The issue of the variance being larger than what one gets with full sample inference, and thus, leading to broader confidence intervals, is different than the confidence intervals having correct coverage.
So, it seems to me that it would be desirable to be able to find a tuning method so that what I’m loosely calling confidence intervals would contain the true parameter values the appropriate percentage of the time, at least when the sample is large enough so that asymptotic results should be a good approximation.