This subtle point you bring up reminded me of Jeffrey Blume’s interval testing procedure he calls “second generation p-values”. While I think the procedure deserves a more descriptive name, with “p-*” being too overloaded in stats and math, I think it is another useful way to extract value from frequentist results.
Jeffrey D. Blume, Robert A. Greevy, Valerie F. Welty, Jeffrey R. Smith & William D. Dupont (2019) An Introduction to Second-Generation p-Values, The American Statistician, 73:sup1, 157-167 (link)
I’d strongly prefer a nice clean separation of the actual data, whether it be interval estimates, likelihood functions, or “confidence” distributions, from the storytelling in the discussion and methods.
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