No. I don't think they would have briefed them on a battle between Klingons and Romulans yet. It had nothing to do with their Vulcan rescue mission, as far as anybody knew. And their rescue mission was an emergency.
They weren't briefed on a battle that took place like 1 or 2 days before, of a battle of 1 Romulan ship against 47 Klingon ships, IN the Neutral Zone, but were immediately briefed on some earthquakes on Vulcan?
I can't agree with that logic.
I also wonder why no one at Starfleet realized that this huge ass ship that just pulverized a whole Klingon fleet entered Federation core space. Apparently, it was sitting somewhere in the Vulcan solar system waiting for the White Hole to open up. And nobody noticed it sitting there, let alone entering Federation space?
But it rationalizes Kirk's rapid promotion, when the rest of Starfleet is depicted as a bunch of totally incompetent twats, he must have been the only one left to command the flagship. Problem is he also won't be briefed about important stuff anymore, and he won't be hiding under Uhura's bed anymore.