A function that measures the distance between two different classes of 'things' (distribution and point, in this case) is necessarily not a distance metric. It

A function that measures the distance between two different classes of 'things' (distribution and point, in this case) is necessarily not a distance metric. It trivially fails to satisfy the triangle inequality, because at least one of d(x,y), d(x,z), d(y,z) will be undefined--no matter how you choose x, y, z you'll end up either trying to measure the distance between two points or the distance between two distributions, neither of which can be handled. This is not a red herring, it's a fundamental issue.