Been a while since I watched First Contact, but I thought they did more than just kill the Queen. Could be wrong, though.

On Section 31, that's the current story, but DS9 had their existence so secret that it's implausible in combination with how open they are in DSC. It'd be like the US disbanding the marine corps and then trying to pretend they never had one. ENT's bit where they quoted the charter also has them pretty clearly deliberately misinterpreting what Article 14, Section 31 actually says.

Harris: "Re-read the charter: Article 14, Section 31. There are a few lines that make allowances for bending the rules during times of extraordinary threat." Archer: "What threat?" Harris: "Take your pick. Earth's got a lot of enemies."

That sounds like it's intended to be meant to cover the wacky reality destroying shenanigans that Starfleet crews tend to get mixed up in. Breaking time travel laws to retrieve whales from the past so that Earth doesn't get wiped off the map, for instance would be covered under art. XIV, §31. Or interfering with the internal politics of the Q Continuum because they're blowing up random stars. Or landing a strike team on a pre-warp planet because they're messing around with Omega Particles. Messing with Romulan internal politics because they hate us doesn't qualify because the Romulans aren't about to wipe anyone out. Romulans are an ordinary threat, not an extraordinary one. That's just Tuesday in the Federation. Harris' quote also doesn't imply that the charter actually calls for the creation of an entire branch of Starfleet to handle such threats.

Before DSC gave them badges, the logical conclusion was that S31 in the prime timeline was an illegal conspiracy among Starfleet officers, not an official organization in any way. Even Into Darkness' portrayal could potentially be interpreted as a S31 conspirator in that timeline getting high enough ranked to move lots of resources on the sly, though I'll admit that's stretching things.