This seems like a satisfactory reduction. By focusing on the analogy with Ultimatum Game, we can capture both the idea of cooperating/defecting, and the idea of evaluating fairness/Shelling point, in a way that follows from the structure of the game, not via references to the informal concepts.

(In my previous attempt to unpack "blackmail", the suggestion was that blackmail is the feeling of the more vague "shouldn't cooperate", as an attempt to get away from the annoying passing-the-buck intuition that it has something to do with the relation of the decision to the status quo or Schelling points (which seem about as confusing as "blackmail" itself). The ultimatum game solution seems to address both concerns, making "defection" less vague and retaining motivation for the "status quo" part of the puzzle.)