I agree. I think the take-home message is that (strictly speaking) frequentist confidence intervals show the compatibility between the data and the model-based expectation along the data-space axis in which the statistic used to calculate the confidence intervals varies. If the model is barely compatible with the data then the confidence intervals will be narrow regardless of the precision of the data. Although this can happen, it is an uncommon scenario (at least in oncology) and we can get away in practice by pretending that the confidence interval width only measures precision.
I can see though why some Bayesians would scoff at this inferential incoherence.