I was surprised not so much at the mention of Section 31, but at the admiral talking about it quite as openly as he did. Perhaps he didn't think it was a problem because he didn't intend Kirk and Spock to come back from their mission, but still, if Section 31 is supposed to be the ultra-secretest of secret organizations, you don't just blurt out that they exist where anyone can overhear you. It makes me think the writers tossed it in so we'd be happy with a DS9 reference without really understanding what it is that makes Section 31 what it is.
BTW, are we supposed to assume that the admiral was actually a member of Section 31 himself? He mostly came off as your average Insane Admiral TM, but actively trying to start a war because you believe it's inevitable but you've got genetically engineered supermen on your side so you'll win anyway does kind of sound like a S31 thing. And the secret base and secret starship with its non-official Starfleet uniforms also fits.