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.
I got the impression that post-Vulcan he'd turned some areas of Section 31 into his own little fiefdom to prepare for the oncoming war, since they shared the same goals, roughly.
I'd argue that ID is still a Section 31 film, but a S31 film shared with 'John Harrison'. Lets be honest, Kirk's trapped between the Machinations of the two, and it's only because its the Enterprise that neither win.