I thought it was pretty clear that Admiral Cornwell was more responsible for his remaining in command, despite the fact that he had changed.
Indeed. And it was also Cornwell who greenlit Georgiou's attack plan and Cornwell who introduced Mirror Georgiou to Discovery's crew, trying to pass her off as the original.
What does THAT tell you?
I could buy Baskin Robbins keeping her in the dark about who and what Lorca really was (given their past together), but she's too high up in the admiralty not to at least know about them, let alone have dealings with them from time to time, especially in the desperation of the Klingon War. And given that the black badges probably
are the ones who got Burnham out of prison in the first place on Lorca's insistence, Cornwell's warning "Don't make any more enemies!" has a whole new dimension of meaning right there.
I don't like that idea. It suggests one of two things, neither of which I like: either (A) Section 31 was somehow privy to important intel (i.e., Lorca's status) that escaped the attention of Starfleet and other Federation authorities, suggesting they're actually really good at what they do, or (B) Starfleet and/or the Federation also knew about Lorca's status, and let him keep his command anyway, suggesting that they're comfortable sanctioning that sort of thing and that Section 31 found out through internal channels.
Or (C) Section 31 found out about Lorca, recruited him to build the spore drive, and then hid his true nature from the rest of Starfleet. That would nicely explain how it is that Lorca managed to bump the learning curve and fit in with our non-barbarian universe; in fact, his monolog when they were Mirrofying the Discovery is probably just him repeating what Leland told him when they were trying to get him to fit in with Starfleet. "Listen, Gabriel. In order to crash a party, you have to look like you
belong there..."
Moreover, either option suggests that Lorca was uncharacteristically incompetent at keeping his secret.
Which is probably why he ended up destroying the Buran with all hands. His crew figured out he was an imposter pretty much from the get-go, but couldn't get their original back, so Lorca killed them all. When Section 31 found him, they figured out what was happening and cut a deal.
The notion that Section 31 has any say whatsoever in who gets the captaincy of important Starfleet ships is not compatible with my interpretation of how Starfleet works. It's a rogue organization, not part of the power structure.
Power structures are controlled by PEOPLE, whose motivations and sympathies vary wildly and are subject to pressure, manipulation, and even blackmail. Section 31 is good at what it does.
I concur with those who have posted that it is not in any formal sense part of the Federation intelligence apparatus.
Sure. It's an unofficial department within Starfleet that is probably at the beck and call of at least one of its senior officers (Admiral Alex Marcus, let's say) who keeps it officially off the books but still directs personnel and funding to it through a very liberal interpretation of Starfleet regulations.