The blog post talks about inference, not prediction, so I find it odd you keep bringing up prediction tasks. There are interesting questions and differences here, but it is very much not the subject of the post.

A standard frequentist tool for making predictions is the prediction interval. This is the appropriate comparison point for Bayesian prediction methods, and exactly the same issues arise as in the comparison of confidence intervals to credible intervals (or posteriors). Namely, frequentist prediction intervals have guaranteed error control, while Bayesian predictions generally do not. So in certain cases you have to choose between being right most of time about your predictions, and being Bayesian.