Were Clopper & Pearson (1934) too careful?

The ‘exact’ interval due to Clopper & Pearson (1934) is often considered to be the gold standard for estimating the binomial parameter. However, for practical purposes it is also often considered to be too conservative, when mean rather than minimum coverage close to nominal could be more appropriate. It is argued that (1) Clopper & Pearson themselves changed between these two criteria, (2) ‘approximate’ intervals are preferable to ‘exact’ intervals, and (3) approximate intervals are well represented by Bayesian intervals based on a uniform prior.