Maximum-support Nash equilibria in zero-sum games
I am not sure about what exactly you mean by the maximum support NE but you may want to check the concept of Correlated Equilibrium. Any convex combination of NE payoff profiles is a Correlated Equilibrium (but not vice versa in general). So, what you wrote made me think about that. However, I don't think it makes much sense to use correlated equilibrium since, you know, there is no obvious advantage of correlation. Also, the NE payoff profile must be constant in your environment.

Another thing that comes to mind is restricting the attention to super-modular games. Then, you get a nice lattice of NE. So, I don't know if this makes sense either.

I would suggest playing around with some examples to understand what exactly it is that you want to characterize and asking a new one based on your examples.