I always took the events of First Contact to be a predestination paradox. The events of First Contact were always meant to happen, and the Enterprise-E was always meant to go back in time, because without their intervention Cochrane's and Lily's mindset towards what they've created wouldn't have been right to develop a relationship with the Vulcans.
Cochrane is shown to be a drunk who doesn't really care much for warp drive beyond it possibly being valuable. Picard with Lilly, and Riker with Cochrane, bring them around to seeing warp drive can be so much more for humanity.
Since it's 7 months after it's airing, I think I'm ok posting about this ... But I thought the Discovery finale opened a can of worms with the Kovich/Daniels reveal, where if you wanted to make him a Section 31 time agent, it might make sense. If he's Daniels and a time agent, why wouldn't he try to alter the timeline to avert the Burn? Either the Burn can't be averted in any timeline, or it serves some sort of agenda Daniels/Kovich believes in.
Plus, does the fact he's going by another name in the 32nd century have any significance?
I could see a story where Daniels is Section 31 (e.g., Daniels states explicitly in Enterprise that he's not a Starfleet officer) and he's considered a "war criminal" after the time war. Elements within the Federation covered up his identity and altered him to protect him from being investigated and prosecuted. And Kovich/Daniels has concentrated enough power that he's powerful enough to direct Federation policy along a path that he believes is best.
i actually don't believe in predestination paradoxes, and think that there is always a first iteration before a loop starts, even if it is replaced and lost to time. However, even if that wasn't the case, I would argue FC can not be one - they were in the future, and saw the timeline change around them; they didn't go to the past, and and up in a situation that causes the reason for them to be there. I don't believe that it applies.
It solves things like first ship named Enterprise; the new NX gets the name because the new timeline knows the name/saw the Enterprise. It also leaves the Borg drones in the past for ENT to find, and gives a reason for ENT having the time war to begin with. It explains every single disconnect in continuity from original TOS/TNG to modern Trek. Then Daniel's and the Borg drone introduce more new technology, and now we have a new 23rd century.
The timewar happened instantly on a 4D scale during the events of First Contact. by the time they return to the future, most things have already been course corrected. This is what the Romulans on SNW are talking about, but they might not know it is all Section 31/Time Cops doing.
I think the Dept of Temporal Investigations morphs into or merges with Section 31 during the TCW.
It would be one of their own ships that survives the initial events of FC and fights the TCW, for Picard to have a home to return to.
Some group/race that came to power in the absence of the Federation would be one of many fronts of the timewar; they would be the heroes of their own stories, trying to keep their world alive.
I'm not convinced that Cochrane wouldn't have gotten around to trying it anyways in the original timeline.
all timecops are war criminals after time travel is outlawed, of course, but if he was also a section 31 agent, they could all be considered war criminals for the things they did/are doing to lead them to that point in rebuilding the timeline.