for whatever reason Morrison thinks that is necessary
I can't speculate as to Morrison's reasoning, but "Checkmate" says to me, "spy agency with its own concerns, probably trying to do mostly the right thing with a certain amount of self-serving agenda and an MO fairly dependent on who's in charge," while "Global Peace Agency" sort of screams "hoshit it's Big Brother these guys will do HORRIBLE THINGS for YOUR OWN GOOD." The first is asking for a sensible, evenhanded treatment. The second is begging to become a boilerplate creepy, evil spider-in-the-shadows. You might have been able to get away with a cheerfully forthright statement of benevolence like putting "global peace" in your actual name back in the Silver Age, but in the modern age it's begging for trouble. So at a guess, Morrison was either trying for that Silver Age simplicity, or more likely, setting up for an oh-so-clever subversion.