You are correct that the "easy interpretation" is not valid for confidence intervals. However, it is the correct interpretation for Bayesian credible intervals.

You are correct that the "easy interpretation" is not valid for confidence intervals. However, it is the correct interpretation for Bayesian credible intervals. Re-read my comment. I think I'm pretty clear that you need to use a method to convert confidence intervals into probability / credible intervals to make them useful. You can treat many frequentist statistical methods as shortcuts to Bayesian methods with flat priors, thus allowing you to exchange confidence (a meaningless concept in practice) for probability. You demonstrated in your simulation what confidence means, but in applications, we only have one confidence interval per parameter, not 1,000,000, so the "reliability of a procedure" interpretation is not useful.