What do you think of broadening the discussion from a single interval to the set of “confidence” intervals, leading to confidence distributions?
I agree that the Bayesian perspective is more natural and relevant, but frequentist reports are common, and I think it is important to understand how to interpret them on their own terms.
Confidence distributions, accoring to Pawitan, have an interpretation as an “extended likelihood” ie. a likelihood that partially accounts for its uncertainty. A more precise definition is given in the paper.
Pawitan, Y., & Lee, Y. (2021). Confidence as likelihood. Statistical Science, 36(4), 509-517. (PDF)