If we work on the assumption that the events of FC
are a predestination paradox, and that the Borg/TNG crews involvement in events there always
was the case..... then the only explanation I can come up with is that Section 31 kept them a bit of a secret, to the point of, for example, the 1701-D computers having no record of them in Q Who. The Hansens figured something out and went out there off their own bat to try and find if these mythical cyborg creatures were true, but it wasn't until Q let the genie out of the bottle that the Borg became "known" to the wider population (Section 31 would, of course, simply pretend they're learning about them at the same time everyone else is).
Doesn't account for
all the discrepencies, but it does tie up the bigger knots.
(I do myself lean more towards the alternate timeline theory, just because I'm a fan of temporal hijinks lol.

)
In regards to the ENT controversies, I am of the view as stated above that maybe the NX ships still existed in the original timeline, maybe even Archer commanded one, but that timeline isn't the version we actually
see in Enterprise. I do like
Unspeakable's explanation above that maybe the original timeline warp five ship wasn't actually called Enterprise in the TOS/TNG version of the timeline, that might be one of the knock on effects from the incursion seen in First Contact. As far as things like the ship wall in TMP and all the "fifth starship to bear the name" stuff in TNG were concerned, the original Warp 5 ship was not an Enterprise, it was christened under a different name.