The new name is just another red herring from FFA. They seriously wouldn't think that calling the team NZ Phoenix is going to result in a massive spike in metrics. Apart from a few fanatics here and there, there would be very few genuine NZ football fans who refuse to support a team called Wellington, and changing that name to NZ is not going to make them become fans all of a sudden.
I suspect the Warriors and Breakers dropping any Auckland reference is more because of the anti-Auckland feeling in the provinces