The Black Badges thing, which have been openly worn on Discovery before, under Lorca (I think only in a very early episode though, they were guarding laboratory or something), suggests a more open organisation than anything in previous Trek - though as you say I don’t think most people make the connection to Section 31. In ENT they seem a bit more covert, and by DS9 they’re not even officially Star Fleet (so not wearing Star Fleet badges), I’d say. It doesn’t violate canon, but it suggests a “rise and fall”. It has been established in canon that they originate with Article 14 (mentioned in this DISCO episode again, actually), Section 31 (hence the name) of the Star Fleet charter. Now that was written in either the 2120s or 2130s.
Now what’s interesting here is - that was when Star Fleet was a human-only organisation (they may have had a few aliens, but it was on operating on behalf of United Earth, not the Federation), and that’s when Section 31 dates back from. And Tyler is, I believe, the first alien we’ve seen in S31. I’m sure he won’t be the last and I’m sure they use alien agents, but my feeling is that they are an organisation working on behalf of humanity, rather than really the Federation (despite the prevarications of Luther Sloan).
Exactly - it can’t last. And as I said, the writers of DISCO are pretty meticulous about canon. They’re definitely aware that by the DS9 era, S31 is a whispered rumour, not an official part of Star Fleet, trumping around with fancy black badges. And I suspect it never has been - they’re named after a paragraph, it’s almost undoubtedly not an official name. Presumably most of the people involved technically have other jobs (like Reed), or aren’t members of Star Fleet (Sloan had a whole fake persona and IIRC claimed to have a family who didn’t know he was in S31 and so on). Plus the complete lack of mention in TOS and TNG means they must have been pretty low-profile for a long time.
I imagine we’ll see Emperor Georgiou take them to a position of great power, overreach, and be forced very far back into the shadows. Either that or the S31 series will get cancelled after a season or two and we’ll never know!