All these confidence interval discussions (both from the original study and from the critique) have no value besides entertaining professors who have more knowl

All these confidence interval discussions (both from the original study and from the critique) have no value besides entertaining professors who have more knowledge of math than common sense. Nothing good will ever come from buzzwords such as "Agresti-Coull 95% interval". Bayesian techniques are not much better since no one will ever agree on the prior. Just treat the study as some super rough point estimate. Adjust for biases such as selection bias if you can. Look at other studies too. Add your personal opinions (e.g., on whether conflicts of interest are relevant here). Complex statistical arguments won't buy you much more than that.