After some discussion (below), I now have a clearer picture of a focused question, so here is a revised question, though some of the comments might now seem unconnected with the original question.
It seems that t-tests converge quickly for symmetric distributions, that the signed-rank test assumes symmetry, and that, for a symmetric distribution, there is no difference between means/pseudomedians/medians. If so, under what circumstances would a relatively inexperienced statistician find the signed-rank test useful, when s/he has both the t-test and sign test available? If one of my (e.g. social science) students is trying to test whether one treatment performs better than another (by some relatively easily interpreted measure, e.g. some notion of "average" difference), I am struggling to find a place for the signed-rank test, even though it seems to generally be taught, and the sign-test ignored, at my university.