what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?

From the same source:

the distribution of the sample mean will be equal to the distribution of the samples themselves; i.e., the sample mean of a large sample is no better (or worse) an estimator of x0 than any single observation from the sample.

Now I'm wondering if there is a symmetric distribution where the sample mean is a strictly worse estimator of the median than a single observation.