I can't see the Vulcans leaving the Earth unsupervised after First Contact.
Even if the surviving Enterprise-E crew did a thorough job of "cleaning up" the mess left behind by themselves (and by the Borg sphere) to avoid alerting the Vulcans that... something odd was going on, leaving a potentially dangerous warp-capable species to its own devices would be too much of a risk - as would leaving the place open for someone else to occupy the region.
Better to establish a de facto protectorate (or trusteeship) over the Sol system, so as to guide the Earth's reconstruction efforts without interference by other warp-capable species - at least until the humans can be trusted enough to handle their own affairs in a manner amenable to long-term Vulcan interests.
Or, at least, so it would have seemed as of 2063. As Ambassador Soval pointed out, it took the Vulcans themselves 1500 years to recover from the Time of Awakening; they were not expecting the Earth to get its own act together as relatively quickly as it did...
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Further, as shown in Regeneration, the Ent-E did not clean things up quite as well as they might have hoped. Indeed, while the "public" face of First Contact would be maintained by both sides, I could well imagine the Vulcan intelligence service being debriefed by Cochrane and by Lily Sloane on what really happened - and for this to lead to a joint Earth-Vulcan entity being tasked with keeping such time travel shenanigans under wraps... and with preparing future generations for the war that is to come.
So while I would not miss the Temporal Cold War being excised from a would-be re-imagining of Enterprise, I would not only keep the episode Regeneration intact; I would lean further into the long-term consequences of the events of Star Trek: First Contact, in terms of how this would lead to the creation of Section 31.
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On another note, this might be a side-effect of my interest in the Star Fleet Universe, but I find myself less keen on the concept of Earth fielding the first "saucer-and-nacelle" ships, prior to the creation of a unified Federation naval service.
Over in the SFU - where the NX-01 does not exist - old Earth ships do not have saucers, though a few (such as the old light cruiser) have familiar-looking warp nacelles. For that matter, none of the other "pre-Star Fleet" ships (Vulcan, Andorians, etc.) in that universe have saucers either. Rather, the first "saucer-and-nacelle" designs are intended from the outset to serve as the symbol of a United Star Fleet, and to mark the evolution of the Federation itself from an alliance into a government.
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To go back to the concept of an Enterprise re-imagining, I would have the Earth ships of the Romulan War be something other than "saucer-and-nacelle" designs, and instead move the onset of the NX-class forward a few decades, to become an early example of a "unified" hull design.
So, there'd be two options for a series:
One would be to set it in the 2150s on an Earth ship named Enterprise, yet not have it be an NX-class.
And two would be to move the story to, say, the 2200s, and to have the first NX-class hull (not named Enterprise) be introduced - as a Federation ship, with a mixed human, Vulcan, Andorian, and Tellarite crew - to mark a "half-step" between the ENT era and the era of the original series.
How does any of that sound?
