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Maybe project the x/y coordinates onto a unit sphere, then find the center of mass in spherical coordinates, then convert those spherical coordinates to the relevant 2d coordinate system you're using, discarding the radius (but keeping it for the information it provides).

That's exactly how it's done, for what it worth (though you use 3D cartesian coordinates instead of spherical coordinates).

Treat each point on the sphere as a 3D unit vector, weight as appropriate, and the center of mass is the sum of all of the vectors.

One way would be to search for points on the sphere which have the closest (weighted) average distance to all the points. Or some similar metric like that.

So instead of finding a point in the middle of the Earth, it would find a point that is geographically closest to the US and China.

This is equivalent to what The Economist did. Average of the cartesian points, extend along the radius to the surface.