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Unfortunately it is popular among those that don't understand or trust statistics. Once had it quoted to me when I gave a scientist an estimate of the number of hours it would take me to design and analyze his experiment. He didn't seem to appreciate the fact that noise can look like effects, and effects of interest can be hidden from intuition by noise.
Edward Tufte is a statistician. Started his career with BA and MS in statistics from Stanford, taught and wrote books about statistics for political scientists and is a fellow of the ASA.
@bart - I don't think you'll run into convergence problems, but you're right that it takes awhile to learn the methods. As for what's wrong with Vb/Va, I'm not sure how to answer because it's not clear to me how those values are being calculated. But under the nested Gaussian assumption described, one good way to define 'best' is in terms of minimizing mean squared error (MSE). This is what the BLUP equations do. See, for example, here.
Nice find, but when Simon writes "view that single observation as a draw from a true probability distribution who's average you want...and you can view that true average itself as having been drawn from a probability distribution of averages" he is describing a mixed or multilevel model. I don't think there's a need to guess that "K=25 seems to work well" because the best linear unbiased predictor equations were worked out more than 60 years ago (BLUP). Excellent Bayesian estimators exist as well. As Brad Efron said, "Those who ignore Statistics are condemned to reinvent it"
