###Does that mean that the only "practical" advantage of the confidence
interval is judging the precision of the study estimate?###
Advantage compared to what?
Leaving aside the issues with what p-values mean, the big advantage I see to interval estimates above hypothesis testing is the direct connection back to the original data. That is presumably what you care about, no?
###The above
discussion suggests that it would be inaccurate to hint that the
study's confidence interval brackets the population estimate.###
Well it depends on what theory of probability you suscribe to.
If you want to make probability statements about parameters as opposed to samples or functions of samples, i.e., estimators---and I would argue you probably do, but you may disagree---you are going to be going Bayesian. This is because in frequentist statistics the locus of randomness is on (usually hypothetical) repeated sampling from a population. Parameters are not random variables. *Estimators* are because they are functions of the data alone
The quote from L. J. Savage: To make the Bayesian omelette, you have to break the Bayesian eggs.
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