Firstly, I would argue that there's no problem in the original analysis. The thought experiment that the author advances says that the spherical mean of the US/China would lie on the north pole. But why is this problematic? What would be the "correct" location? Furthermore, in such a bimodal distribution, all measures of centrality are going to be located in low probability areas. This isn't a problem with the original Economist infographic so much as a limitation of an average.
Secondly, the "fix" that the author has advanced makes a very serious errors: the choice of longitude to split at has a huge influence on the center of mass. The author has chosen the 180th meridian (by naïvely using longitude as if it were not cyclic, it seems) which privileges Europe in this analysis (as the choice of prime meridian privileges Europe historically, amusingly). Moving the split so that (for example) the east and west halves of the US were on opposite ends of the map would effectively negate the contribute of the US to the calculation, which is an inherent problem with using the longitude in this way.
