Funnily enough, I think neither of those notions are the correct thing to optimize. For me, you want to minimize the maximum number of guesses. This leads to an equilibrium. Otherwise, your opponent can just abuse your strategy.
I don't think a maximum-number-of-guesses optimizer leads to an equilibrium (if you mean Nash equilibrium), assuming you're playing the game where the score of the game for the setter is the number of guesses. In particular, if the solver's strategy is deterministic, the setter will always pick one of the words that needs 5 guesses. There's no reason to think the nash equilibrium can be easily found.
If you don't care about optimizing the EV at all, the greedy tree already has the property you're interested in.
The entity choosing the word in a game like Wordle or Hang Man is more like the dungeon master in a game of D&D than an opponent in the traditional sense? They are there to help you have fun.