How I view the press conferences is that KJ has a plan to announce each step in his process separately. The local investors, the state support and the whale. This last press conference was about the state support in the process, which happened to coincide with the day that Stern and Burkle met. The whale will be announced when it's time.
As far as the meeting with Stern and Burkle, both men are rich/powerful and busy. You meet face to face when you are finalizing or coming to an agreement on something. They wouldn't waste each others time with a 2 hour face to face meeting that the NBA themselves acknowledged took place. This public acknowledgment by the NBA is important, as it is not a mistake. The information wasn't "leaked." If Burkle did not want it to be known he was meeting with the NBA, we wouldn't know. If the meeting was negative, Burkle walks away and KJ looks to Ellison or someone else, all behind closed doors.