I guess a simpler explanation might be that the DISCOVERY version of Section 31 is totally destroyed, except the absence of Section 31 simply causes new clandestine agents to take on black ops work on behalf of the Federation as per Article 31 of the Federation Charter and use the Section 31 name for a new organization that has no official ties to Starfleet and operates independently, becoming the Section 31 we see in DEEP SPACE NINE. Again, if Section 31 ceased to exist, someone else would simply create it.
That would work.
I guess if you had to retroactively have the changes make sense, in TOS, time-travel becomes a regular and easy to do thing and perhaps, concerned over temporal espionage, the heads of Section 31 came to the (right?) conclusion that they had to hunker down and operate much, much more covertly and leaving few files or evidence for time-travelling saboteurs in the future to use against them in (their) present.
Also, they may have had big reservations with having to share info with the Klingon Empire, once they were at peace with Starfleet and eventually becoming a part of it.