Calling this 'alternative' construction seems like coming full circle since this line of combinatorial argument is how Boltzmann came up with his H-function in the first place, which inspired Shannon's entropy.
Yep! This relationship is well known in statistical mechanics. I was just surprised that in many years of intersecting with information theory in other fields (computational neuroscience in particular) I'd never come across it before, even though IMO it provides an insightful perspective.