Yeah we've seen no evidence that First Contact really changed anything and plenty of evidence (Deep Space Nine, Voyager, In a Mirror Darkly, These are the Voyages...) that the timeline is just as we saw it. We definitely got no evidence or even hints that First Contact caused the Temporal Cold War.
Well I, for one, can't think of any better reason for it to suddenly pop into existance. YMMV of course.
There have been more than a handful of ways that post FC lore has completely contradicted TOS and original TNG lore, and this is a very simple explanation, that is already there in the narrative, that explains literally anything that can be complained about.
IMO, the NX01 was only named ENTERPRISE because of First Contact, and it was saved from destruction to repurpose its mission to ensure the founding of the Federation after the events of F.C. completely screwed up the future. This all happened, 4th dimensionally, in the blink of an eye, in the instance before the "E" returns to its present, by the time of which things have mostly course corrected, with the 22nd and 23rd centuries taking the brunt of the butterflies.