Good post. I wonder where you got the motivation for these maunderings?!

It is probably worth pointing out that if the moment estimates are infeasible, there is a strong chance (ahem, high likelihood) that the distribution is not appropriate for the observed data. Thus, computing the moment estimates can serve a dual purpose: 1) provide some rough indication of whether the distributional assumptions are reasonable for the observed data, and 2) produce initial estimates for a likelihood estimation procedure.