A 95% confidence interval is described as having a 95% chance of containing the true value. Why does a statistician object, and what is the correct reading?

C. Because the parameter is fixed; the 95% describes the procedure

Nothing is random once the interval exists, the sampling was random, so the guarantee attaches to the method rather than to this interval. The reading people want is available from a Bayesian credible interval, at the price of specifying a prior, which is exactly what the frequentist approach declines to do.