The thing that I was vaguely thinking about taking it further is that not every topological sort of that graph (or, if you prefer poset terminology, not every total order consistent with that partial order) represents a fair ranking system. For instance, you could choose to rank a country with two silvers above one with a single gold, but further up the table you might choose to rank a country with one gold and a bunch of other stuff above a country with two silvers and the same other stuff. Locally, each decision would be consistent with the partial order, but globally one has to look at it and say there's something fishy about both decisions occurring that way in the same table.
So one might introduce a "fairness" or "consistency" criterion along the lines of insisting that this sort of decision has to go the same way throughout the table. Formally, it would go something along the lines of: suppose you have some function which maps a difference tuple of gold, silver and bronze medals to {positive,negative,zero}, in such a way that it's consistent with the previous rules (non-negative whenever g,g+s,g+s+b are all non-negative, non-positive when they're all non-positive) and also consistent with obvious ordering constraints (e.g. adding two positive things gives a positive thing). Then a total order can be derived from such a function by ranking each pair of countries according to the vector difference of their medal counts. Any such total order is "fair", in that there's some consistently applied comparison rule which gives rise to it; a total order which can be generated by no function in the above way is unfair and we want to discount it.
However, it wouldn't change the Hasse diagram as shown. I've convinced myself that any pair of countries marked as incomparable on my diagram can be ranked in an order of your choice by choosing an appropriate additive scoring system (x points for a bronze, y > x for a silver, z > y for a gold), and ranked in the other order by a different one. Any such additive scoring system is fair by the above definition, so no additional pair of countries becomes comparable due to the introduction of a fairness requirement.
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So one might introduce a "fairness" or "consistency" criterion along the lines of insisting that this sort of decision has to go the same way throughout the table. Formally, it would go something along the lines of: suppose you have some function which maps a difference tuple of gold, silver and bronze medals to {positive,negative,zero}, in such a way that it's consistent with the previous rules (non-negative whenever g,g+s,g+s+b are all non-negative, non-positive when they're all non-positive) and also consistent with obvious ordering constraints (e.g. adding two positive things gives a positive thing). Then a total order can be derived from such a function by ranking each pair of countries according to the vector difference of their medal counts. Any such total order is "fair", in that there's some consistently applied comparison rule which gives rise to it; a total order which can be generated by no function in the above way is unfair and we want to discount it.
However, it wouldn't change the Hasse diagram as shown. I've convinced myself that any pair of countries marked as incomparable on my diagram can be ranked in an order of your choice by choosing an appropriate additive scoring system (x points for a bronze, y > x for a silver, z > y for a gold), and ranked in the other order by a different one. Any such additive scoring system is fair by the above definition, so no additional pair of countries becomes comparable due to the introduction of a fairness requirement.