I think it's easily explained away that Section 31 eventually discovered the need to be more secretive, and conceal their presence even from Starfleet.They've definitely based their version of Section 31 on the Kelvinverse one. They have their own black ships loaded with cool tech, their own uniforms and they're basically Starfleet's Intelligence and black ops division. And everyone knows who they are.
Compare to DS9, where nobody knew what section 31 was, they used existing Starfleet ships (the Bellerophon) and explicitly had no headquarters of any kind.