"It's one of the first examples you hit in many textbooks"

Yes! Thank you! I am NOT disputing Bayesian analysis or the textbooks - rather, a group of NON-stat folks in Berkeley have decided they can "just plug numbers into Bayes' Theorem by itself, with nothing else, and feel confident...without a confidence interval." THOSE folks need to read a textbook, because "Bayes' Theorem alone" is out-dated by the improvements you mentioned!