And by the next use of the Sling Shot (Assignment Earth) it was another day at the office.
Captain's log. Using the lightspeed breakaway factor, the Enterprise has moved back through time to the twentieth century. We are now in extended orbit around Earth, using our ship's deflector shields to remain unobserved. Our mission, historical research. We are monitoring Earth communications to find out how our planet survived desperate problems in the year 1968.

^ I admit, we don't know if he did or not. But Archer was pissed enough at Daniels for continually dragging him into it that I suspect he just washed his hands of the whole thing.
Especially since Daniels was so far in the future that Archer couldn't possibly have had much to mention in the first place. All Archer knew was, Daniels could transport him through time with a thought. That's not the kind of thing you can keep detailed documentation about. Somebody that powerful, probably best to let it go and forget all about it.
I think that would've have been rather difficult for him to pull off, as all the bridge staff knew about it and came unequivocally to believe it, even T'Pol, as would just about the entire crew, one would have to imagine, given what they directly experienced. How would he have gotten everyone to keep quiet about it? Taken a blood oath?
Kirk seems to have redacted things like Cochrane and Khan's locations and Gary Mitchell's "ascension".
OK, I'll bite on the first two, but I don't know where you're getting any substantiation for Mitchell's case. I don't recall any canon citation that such happened. Why would he have done so? I might argue elsewhere, that he could have to have hidden his culpability in the incident, but I doubt that's what you mean. Mitchell was cited for dying in support of doing his duty, though I suppose the exact circumstances might have been obscured. But if for no other reason than making it known to Starfleet the dangers of encountering the Barrier, I can't imagine Kirk being so irresponsibly lax and capricious. Again, what woud be the purpose?
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