So, is your point that we often don't know when a purportedly 95% confidence interval really is one?

I'm saying that this stuff about 95% CI is a completely empty and broken promise; if we see the coverage blown routinely, as we do in particle physics in this specific case, the CI is completely useless - it didn't deliver what it was deductively promised. It's like have a Ouija board which is guaranteed to be right 95% of the time, but oh wait, it was right just 90% of the time so I guess it wasn't really a Oujia board after all.

Even if we had this chimerical '95% confidence interval', we could never know that it was a genuine 95% confidence interval. I am reminded of Borges:

It is universally admitted that the unicorn is a supernatural being of good omen; such is declared in all the odes, annals, biographies of illustrious men and other texts whose authority is unquestionable. Even children and village women know that the unicorn constitutes a favorable presage. But this animal does not figure among the domestic beasts, it is not always easy to find, it does not lend itself to classification. It is not like the horse or the bull, the wolf or the deer. In such conditions, we could be face to face with a unicorn and not know for certain what it was. We know that such and such an animal with a mane is a horse and that such and such an animal with horns is a bull. But we do not know what the unicorn is like.

It is universally admitted that the 95% confidence interval is a result of good coverage; such is declared in all the papers, textbooks, biographies of illustrious statisticians and other texts whose authority is unquestionable...

(Given that "95% CIs" are not 95% CIs, I will content myself with honest credible intervals, which at least are what they pretend to be.)